Coherent sheaves, superconnections, and RRG
CCOHERENT SHEAVES, SUPERCONNECTIONS, ANDRRG
JEAN-MICHEL
BISMUT , SHU
SHEN , AND ZHAOTING
WEIAbstract.
Given a compact complex manifold, the purpose ofthis paper is to construct the Chern character for coherent sheaveswith values in Bott-Chern cohomology, and to prove a correspond-ing Riemann-Roch-Grothendieck formula. Our paper is based ona fundamental construction of Block.
Contents
1. Introduction 41.1. The construction of ch BC bcoh ( X ) 153.1. Definition of the derived category D bcoh ( X ) 163.2. Pull-backs 163.3. Tensor products 173.4. Direct images 174. Preliminaries on linear algebra and differential geometry 174.1. Filtered vector space and exterior algebras 184.2. Supercommutators and supertraces 214.3. Morphisms and cones 224.4. Generalized Hermitian metrics 234.5. Clifford algebras 24 Mathematics Subject Classification.
Key words and phrases.
Derived categories, Riemann-Roch theorems, Cherncharacters, Hypoelliptic equations. a r X i v : . [ m a t h . AG ] F e b JEAN-MICHEL
BISMUT , SHU
SHEN , AND ZHAOTING
WEI
H E = 0 365.6. Superconnections, morphisms, and cones 385.7. Pull-backs 395.8. Tensor products 406. An equivalence of categories 406.1. Some properties of D bcoh ( X ) 416.2. Three categories 426.3. Essential surjectivity 436.4. The homotopy category 526.5. An equivalence of categories 526.6. Compatibility with pull-backs 536.7. Compatibility with tensor products 546.8. Direct images 557. Antiholomorphic superconnections and generalized metrics 567.1. The adjoint of an antiholomorphic superconnection 577.2. Curvature 608. Generalized metrics and Chern character forms 608.1. The Chern character forms 618.2. A trivial example 658.3. The Chern character of pull-backs 658.4. Chern character and tensor products 668.5. The case where H E is locally free 668.6. The Chern character form and the scaling of the metric 678.7. The Chern character of a cone 708.8. The Chern character on D bcoh ( X ) 708.9. The Chern character on K ( X ) 718.10. Spectral truncations 739. The case of embeddings 779.1. Embeddings, direct images, and transversality 779.2. Deformation to the normal cone 799.3. A Riemann-Roch-Grothendieck theorem for embeddings 819.4. The uniqueness of the Chern character 8410. Submersions and elliptic superconnections 8510.1. A theorem of Riemann-Roch-Grothendieck forsubmersions 8610.2. Replacing F by E OHERENT SHEAVES AND RRG 3 A p ∗ E (cid:48)(cid:48) T X A D , A p ∗ E , H Rp ∗ E is locally free 10412. A proof of Theorem 10.1 when ∂ X ∂ X ω X = 0. 10413. The hypoelliptic superconnections 11013.1. The total space of (cid:100) T X (cid:100)
T X E M A E M (cid:48)(cid:48) A (cid:48)(cid:48) A E M (cid:48)(cid:48) z A Z A Z , B Z A Z g (cid:100) T X b → ∂ X ∂ X ω X = 0 13115.1. Finite and infinite-dimensional traces 13115.2. The time parameter 13415.3. The case when ∂ X ∂ X ω X = 0 13716. Exotic superconnections and Riemann-Roch-Grothendieck 14316.1. A deformation of the K¨ahler form ω X A Z,θ A Y,θ
JEAN-MICHEL
BISMUT , SHU
SHEN , AND ZHAOTING
WEI t → (cid:16) A (cid:48)(cid:48) Y , ω X /t, g D , g (cid:100) T X /t , θ t (cid:17) Introduction
Let X be a compact complex manifold. Let K · ( X ) , K · ( X ) be theGrothendieck groups of the locally free O X -sheaves, and of the O X -coherent sheaves on X , and let H • , • BC ( X, R ) be the Bott-Chern coho-mology of X . Set(1.1) H (=)BC ( X, R ) = dim X (cid:77) p =0 H p,p BC ( X, R ) . There are characteristic classes K · ( X ) → H (=)BC ( X, R ), like the Cherncharacter ch BC , or the Todd class Td BC , that refine on the classicalChern classes with values in the even de Rham cohomology H even ( X, R ).In the present paper, we extend the definition of the Chern charac-ter ch BC : K · ( X ) → H (=)BC ( X, R ) to a map K · ( X ) → H (=)BC ( X, R ).In degree (1 , Y is another compact complex manifold, let f : X → Y be a holomor-phic map. Let f ! : K · ( X ) → K · ( Y ) be the direct image, and let f ∗ : H (=)BC ( X, R ) → H (=)BC ( Y, R ) denote the push-forward map. Theorem 1.1. If F ∈ K · ( X ) , then (1.2)Td BC ( T Y ) ch BC ( f ! F ) = f ∗ [Td BC ( T X ) ch BC ( F )] in H (=)BC ( Y, R ) . When X is projective, then K · ( X ) = K · ( X ), and our definition ofch BC is the standard one. Similarly, when X, Y are projective, Theorem1.1 is a consequence of Riemann-Roch-Grothendieck.We will describe the ideas and techniques that are used to obtainTheorem 1.1.1.1.
The construction of ch BC . If E is a holomorphic vector bundleon X , and if g E is a Hermitian metric on E , there is an associatedunitary Chern connection, whose curvature is of type (1 , (cid:0) E, g E (cid:1) , which is closed and lies in Ω (=) ( X, R ). Results of Bott OHERENT SHEAVES AND RRG 5 and Chern [BoC65] show that the class of ch (cid:0)
E, g E (cid:1) in H (=)BC ( X, R )does not depend on g E , which gives us a Chern character map ch BC : K · ( X ) → H (=)BC ( X, R ).If X is projective, if E ∈ K · ( X ), it has a finite locally free projectiveresolution R . It can be shown that ch BC ( R ) does not depend on R , sothat we get a Chern character map ch BC : K · ( X ) → H (=)BC ( X, R ). Asshown by Voisin [V02], if X is not projective, such projective resolutionsmay well not exist .To sidestep this problem, we use a fundamental construction of Block[Bl10]. To explain this construction, we replace K · ( X ) by D bcoh ( X ),the derived category of bounded O X -complexes with coherent cohomol-ogy. In [Bl10], Block established an equivalence of categories betweenD bcoh ( X ) and another category B ( X ) of objects which extend holomor-phic vector bundles. More precisely, an object in B ( X ) is representedby E : (cid:0) E, A E (cid:48)(cid:48) (cid:1) , where E a Z -graded vector bundle which is a freeΛ (cid:0) T ∗ X (cid:1) -module, and A E (cid:48)(cid:48) is an antiholomorphic superconnection on E in the sense of Quillen [Qu85] such that A E (cid:48)(cid:48) , = 0.Following earlier work by Qiang [Q16, Q17], as in the case of holo-morphic vector bundles, one can define generalized metrics on E , anduse these to define a Chern character ch BC : B ( X ) → H (=)BC ( X, R ),from which we get a map ch BC : D bcoh ( X ) → H (=)BC ( X, R ), and finallyan associated map ch BC : K · ( X ) → H (=)BC ( X, R ). Also we show that indegree (1 , bcoh ( X ).1.2. The Riemann-Roch theorem.
Let us now describe the mainsteps in our proof of Theorem 1.1. As in Borel-Serre [BorS58, Section7], the proof is reduced to the case where f is an embedding or aprojection. Indeed, let i : H ⊂ X × Y be the graph of f , and let p bethe projection X × Y → Y . Then f = pi . Also X ∼ H , so that i is anembedding X → X × Y . By functoriality, we will reduce the proof tothe case where f = i or f = p .1.2.1. The case of embeddings.
In the case of embeddings, the proofis obtained by a suitable adaptation of the deformation to the normalcone. The proof relies mostly on the functorial properties of the aboveobjects E . In particular, for a generic torus of dimension ≥
3, the ideal sheaf of a pointdoes not admit a finite locally free resolution.
JEAN-MICHEL
BISMUT , SHU
SHEN , AND ZHAOTING
WEI
The case of projections.
In the case of a projection p : M = S × X → S , we reduce the proof to the case where F is representedby an object E in B ( M ). By a theorem of Grauert [GrR84, Theorem10.4.6], p ∗ E is known to define an object in D bcoh ( S ), but it is not anobject in B ( S ), because it is infinite-dimensional.Then, we adapt the methods of [B13], where the case of proper holo-morphic submersions was considered, when E is a holomorphic vec-tor bundle, and the direct image Rp ∗ E is locally free, in which casech BC ( Rp ∗ E ) can be defined without using the theory of Block [Bl10].The proof was obtained using the infinite-dimensional elliptic super-connections on the infinite-dimensional p ∗ E of [B86, BGS88b, BGS88c],their hypoelliptic deformations , as well as local index theory .Given metric data which involve in particular a Hermitian metric on T X with K¨ahler form ω X , we define infinite-dimensional Chern char-acter forms, and we prove that these forms represent ch BC ( Rp ∗ E ). For t >
0, when replacing ω X by ω X /t , if ∂ X ∂ X ω X = 0, we take the limit ofour elliptic superconnection forms, and we prove Theorem 1.1. Whenthere is no such a metric, we inspire ourselves from the methods of[B13], where superconnections with hypoelliptic curvature are defined,which ultimately allow us to prove Theorem 1.1. In the constructionsand in the proofs, the fact we deal with antiholomorphic superconnec-tions instead of holomorphic vector bundles introduces extra difficultieswith respect to [B13].1.2.3. The spectral truncations.
In Subsections 8.10 and 11.1, we de-velop a technique of spectral truncations, which is used when deal-ing with the infinite-dimensional p ∗ E . This method extends classicalspectral truncations to families, while preserving critical cohomologicalinformation.1.3. Earlier work.
Let us now describe earlier related work. In [Gri10],Grivaux defined a Chern character on K · ( X ) with values in the ratio-nal Deligne cohomology ring A ( X ), and he proved Riemann-Roch-Grothendieck for projective morphisms. He also proved a uniquenessresult for characteristic classes on K · ( X ) with values in a cohomologytheory that verifies a simple set of axioms. Recently, following earlierwork by Green [Gre80], Toledo-Tong [TT86], Hosgood [Ho20a, Ho20b]constructed resolutions of coherent sheaves by simplicially locally free This refers to properties of the curvatures of the superconnections, that arefiberwise elliptic differential operators, or fiberwise hypoelliptic operators. Local index theory is a method which computes the local asymptotics of thesupertraces of certain heat kernels as t → OHERENT SHEAVES AND RRG 7 sheaves, and obtained Chern classes of coherent sheaves in de Rhamcohomology (and in twisted de Rham cohomology), which he showedto be compatible with the construction of Grivaux [Gri10].In [W20], Wu has proved a Riemann-Roch-Grothendieck theorem forcoherent sheaves and projective morphisms. His Chern character takesvalues in rational Bott-Chern cohomology. He proves a Riemann-Roch-Grothendieck for projective morphisms, while our results are uncondi-tional.As explained before, the constructions and results of Block [Bl10]play a fundamental role in our paper, as an inspiration, and also forfundamental ideas. Also in [Q16, Q17], Qiang had constructed a Cherncharacter ch BC : K · ( X ) → H (=) ( X, R ) by essentially the same meth-ods as ours. More detailed comparison with his results will be given inSection 8.Moreover, we show that our Chern character ch BC : K · ( X ) → H (=)BC ( X, R ) verifies the uniqueness conditions of Grivaux [Gri10].1.4. The organization of the paper.
This paper is organized asfollows. In Section 2, we recall various properties of Bott-Chern co-homology, and we recall the construction of characteristic classes withvalues in Bott-Chern cohomology.In Section 3, we state various properties of the derived categoryD bcoh ( X ), and of the associated functors.In Section 4, we recall elementary properties of free modules over anexterior algebra.In Section 5, we introduce antiholomorphic superconnections E = (cid:0) E, A E (cid:48)(cid:48) (cid:1) , we show that the corresponding cohomology sheaf H E iscoherent, and that the associated spectral sequence degenerates at E .In particular, we prove the result of Block [Bl10] asserting that locally, E is just a complex of holomorphic vector bundles.In Section 6, we establish the result of Block [Bl10] on the equivalenceof categories B ( X ) (cid:39) D bcoh ( X ).In Section 7, given a splitting E (cid:39) E , with E = Λ (cid:0) T ∗ X (cid:1) (cid:98) ⊗ D ,we define the generalized metrics on E , and the adjoint A E (cid:48) of theantiholomorphic superconnection A E (cid:48)(cid:48) . We also introduce the curva-ture (cid:2) A E (cid:48)(cid:48) , A E (cid:48) (cid:3) . This chapter is the extension of what is done forholomorphic Hermitian vector bundles, when constructing the corre-sponding Chern connection. In the paper, D is called the diagonal bundle associated with E . If A , A (cid:48) are Z -graded algebras, A (cid:98) ⊗A (cid:48) is the Z -graded algebra which is the tensor product of A , A (cid:48) . JEAN-MICHEL
BISMUT , SHU
SHEN , AND ZHAOTING
WEI
In Section 8, given a generalized metric h , we construct the Cherncharacter form ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) ∈ Ω (=) ( X, R ), whose Bott-Chern coho-mology class does not depend on h , or on the splitting and will bedenoted ch BC (cid:0) A E (cid:48)(cid:48) (cid:1) . We show that ch BC induces a map D bcoh ( X ) → H (=)BC ( X, R ), and ultimately a map K · ( X ) → H (=)BC ( X, R ). This Sec-tion gives another approach to earlier results by Qiang [Q16, Q17].In Section 9, we establish Theorem 1.1 when f is an embedding i : X → Y . Using the deformation to the normal cone, we reduce theproof to the embedding X → P (cid:0) N X/Y ⊕ C (cid:1) . Also we prove that ourChern character ch BC : K · ( X ) → H (=)BC ( X, R ) verifies the uniquenessconditions of Grivaux [Gri10].In Section 10, when f is the projection p : M = S × X → S ,and E is an antiholomorphic superconnection on M , we constructthe infinite-dimensional antiholomorphic superconnection (cid:0) p ∗ E , A p ∗ E (cid:48)(cid:48) (cid:1) over S . Given a splitting E (cid:39) E , a metric g T X on T X with K¨ahlerform ω X , and a Hermitian metric g D on D , we construct the L -adjoint A p ∗ E (cid:48) of A p ∗ E (cid:48)(cid:48) , and we give a Lichnerowicz formula for thecurvature (cid:2) A p ∗ E (cid:48)(cid:48) , A p ∗ E (cid:48) (cid:3) . Also we construct the elliptic Chern char-acter forms ch (cid:0) A p ∗ E (cid:48)(cid:48) , ω X , g D (cid:1) , and their common Bott-Chern classch BC (cid:0) A p ∗ E (cid:48)(cid:48) (cid:1) ∈ H (=)BC ( S, R ).In Section 11, we prove that ch BC (cid:0) A p ∗ E (cid:48)(cid:48) (cid:1) = ch BC ( Rp ∗ E ). The proofis based on the construction of a quasi-isomorphism from a classicalantiholomorphic superconnection E to p ∗ E . Spectral truncations playa critical role in the proofs.In Section 12, we prove Theorem 1.1 for the projection p when ∂ X ∂ X ω X = 0. The proof is obtained by taking the limit as t → (cid:0) A p ∗ E (cid:48)(cid:48) , ω X /t, g D (cid:1) by methods of local index theory.In Section 13, we construct superconnections with hypoelliptic cur-vature. More precisely, if (cid:100) T X is another copy of
T X , we introducethe total space M of (cid:91) T R X , and also the tautological Koszul complexassociated with the embedding M → M . Given a splitting E (cid:39) E ,and metrics g T X , g (cid:100)
T X , g D on T X, (cid:100)
T X, D , we construct a nonpositiveHermitian form (cid:15) X . The adjoint of our new antiholomorphic supercon-nection A (cid:48)(cid:48) Y over S is constructed using (cid:15) X . The corresponding curva-ture is a fiberwise hypoelliptic operator. Our constructions combinethe methods of [B13] with the results of the previous Sections.In Section 14, the hypoelliptic superconnection forms ch (cid:16) A (cid:48)(cid:48) Y , ω X , g D , g (cid:100) T X (cid:17) are constructed, and their Bott-Chern class is shown to be the same asthe class of the elliptic superconnection forms.
OHERENT SHEAVES AND RRG 9
In Section 15, when ∂ X ∂ X ω X = 0, we give another proof of Theorem1.1 for p . The results of this Section duplicate the results of Section12, with techniques that prepare for a proof of the Theorem 1.1 in fullgenerality.Finally, in Section 16, we introduce a new deformation of our hy-poelliptic superconnection, in which the K¨ahler form is deformed toan object that is still a (1 , Y , and we prove Theorem 1.1 for p , which completes theproof of this Theorem in full generality.In the whole paper, if A = A + ⊕ A − is a Z -graded algebra, if a, b ∈ A ,we denote by [ a, b ] the supercommutator of a, b , which is bilinear in a, b .More precisely, if a ∈ A , setdeg a =0 if a ∈ A + , (1.3) 1 if a ∈ A − . If a, b ∈ A ± , then(1.4) [ a, b ] = ab − ( − deg a deg b ba. If A is trivially Z -graded, then [ a, b ] is the standard commutator.2. Bott-Chern cohomology and characteristic classes
The purpose of this Section is to recall basic facts on Bott-Cherncohomology of a compact complex manifold X .This Section is organized as follows. In Subsection 2.1, we recall thedefinition of the Bott-Chern cohomology H • , • BC ( X, C ).In Subsection 2.2, we give various properties of the Bott-Chern Lapla-cian of Kodaira-Spencer, which imply that H • , • BC ( X, C ) is finite-dimensional,that it can be represented by smooth forms or by currents, and alsothat it is a bigraded algebra.In Subsection 2.3, we describe various functors that act on Bott-Chern cohomology.Finally, in Subsection 2.4, we explain the construction of character-istic classes of holomorphic vector bundles with values in Bott-Cherncohomology.2.1. The Bott-Chern cohomology.
Let X be a compact complexmanifold of dimension n . Let T X be the holomorphic tangent bundle,and let T R X be the corresponding real tangent bundle. If T C X = T R X ⊗ R C , then T C X = T X ⊕ T X .Let (Ω ( X, R ) , d ) be the de Rham complex of smooth real differentialforms on X . The de Rham cohomology groups H ( X, R ) are defined BISMUT , SHU
SHEN , AND ZHAOTING
WEI by(2.1) H • ( X, R ) = ker d ∩ Ω • ( X, R ) /d Ω •− ( X, R ) . If α ∈ Ω ( X, R ) is closed, let [ α ] ∈ H ( X, R ) be its cohomology class.We may as well replace R by C , and obtain the complexification of theabove objects.We follow [D09, Section 6.8] and [Sc07, Section 1]. For 0 ≤ p, q ≤ n ,let Ω p,q ( X, C ) be the vector space of smooth sections of Λ p,q ( T ∗ C X ) =Λ p ( T ∗ X ) (cid:98) ⊗ Λ q (cid:0) T ∗ X (cid:1) . Note that d splits as d = ∂ + ∂ . Definition 2.1.
The Bott-Chern cohomology groups H p,q BC ( X, C ) aregiven by(2.2) H p,q BC ( X, C ) = (Ω p,q ( X, C ) ∩ ker d ) /∂∂ Ω p − ,q − ( X, C ) . Then H • , • BC ( X, C ) is a bigraded vector space. If α ∈ Ω p,q ( X, C ) isclosed, let { α } be the class of α in H p,q BC ( X, C ). There is a canonicalmap H p,q BC ( X, C ) → H p + q ( X, C ) that maps { α } to [ α ].Then H • , • BC ( X, C ) inherits from Ω • , • ( X, C ) the structure of a bi-graded algebra.PutΩ (=) ( X, C ) = (cid:77) ≤ p ≤ n Ω p,p ( X, C ) , H (=)BC ( X, C ) = (cid:77) ≤ p ≤ n H p,p BC ( X, C ) . (2.3)The vector spaces Ω (=) ( X, C ) , H (=)BC ( X, C ) are preserved by conjuga-tion. We obtain this way the real vector space Ω (=) ( X, R ) and the cor-responding real Bott-Chern cohomology H (=)BC ( X, R ). Also H (=)BC ( X, R )is an algebra.2.2. The Bott-Chern Laplacian of Kodaira-Spencer.
Let g T X bea Hermitian metric on
T X . We equip Ω ( X, C ) with the associated L Hermitian product, so that the Ω • , • ( X, C ) are mutually orthogonal.If A is a differential operator acting on Ω ( X, C ), we denote by A ∗ itsformal adjoint. If A, B are two operators, let [
A, B ] + = AB + BA betheir anticommutator.We introduce the Kodaira-Spencer Laplacian [KoS60, Proposition5], [Sc07, Section 2b],(2.4) (cid:3) X BC = (cid:104) ∂∂, (cid:0) ∂∂ (cid:1) ∗ (cid:105) + + (cid:104) ∂ ∗ ∂, (cid:16) ∂ ∗ ∂ (cid:17) ∗ (cid:105) + + ∂ ∗ ∂ + ∂ ∗ ∂. Then (cid:3) X BC is a formally self-adjoint nonnegative real operator actingon Ω ( X, R ) and preserving the bidegree. This operator is elliptic oforder 4, with scalar principal symbol | ξ | T ∗ R X / OHERENT SHEAVES AND RRG 11
Let D ( X, R ) be the vector space of real currents on X , and let D ( X, C ) be its complexification. Then Ω ( X, R ) ⊂ D ( X, R ). If u ∈ D ( X, C ), if (cid:3) X BC u = 0, since (cid:3) X BC is elliptic, then u ∈ Ω ( X, C ).Set(2.5) H ( X ) = ker (cid:3) X BC . Then H ( X ) is also bigraded, and moreover,(2.6) H ( X ) = ker ∂ ∩ ker ∂ ∩ ker ∂ ∗ ∂ ∗ . By [Sc07, Theorem 2.2], we have the orthogonal splitting(2.7) Ω ( X, C ) = H ( X ) ⊕ im ∂∂ ⊕ (cid:16) im ∂ ∗ + im ∂ ∗ (cid:17) , and we have the canonical isomorphism of bigraded vector spaces,(2.8) H BC ( X, C ) (cid:39) H ( X ) . In particular, the Bott-Chern groups H • , • BC ( X, C ) are finite dimensional.By the above, if we define Bott-Chern cohomology using instead D ( X, C ), it is the same as before, and it is still represented canonicallyby the smooth forms in H ( X ). While, in general, the multiplication ofcurrents is not well-defined, the multiplication of the Bott-Chern classesof currents is still well-defined via the multiplication of correspondingsmooth representatives.If s ∈ ker ∂ ∩ ker ∂ , then(2.9) (cid:3) X BC s = ∂∂ (cid:0) ∂∂ (cid:1) ∗ s, so that if s ∈ Ω ( X, C ), then(2.10) (cid:3) X BC ∂∂s = ∂∂ (cid:0) ∂∂ (cid:1) ∗ ∂∂s. In particular (cid:3) X BC preserves ker ∂ ∩ ker ∂ and im ∂∂ , while mapping thefirst vector space into the second one. By (2.6), we have the orthogonalsplitting,(2.11) ker ∂ ∩ ker ∂ = H ( X ) ⊕ im ∂∂, (cid:3) X BC preserves the splitting in (2.11), and H ( X ) is its kernel.For t >
0, let P t ( x, x (cid:48) ) denote the smooth kernel associated withthe operator exp (cid:0) − t (cid:3) X BC (cid:1) with respect to the Riemannian volume dx (cid:48) .The operator exp (cid:0) − t (cid:3) X BC (cid:1) preserves ker ∂ ∩ ker ∂ .For t >
0, if s ∈ D ( X, C ), then(2.12) exp (cid:0) − t (cid:3) X BC (cid:1) s − s = − (cid:90) t (cid:3) X BC exp (cid:0) − u (cid:3) X BC (cid:1) sdu. BISMUT , SHU
SHEN , AND ZHAOTING
WEI
For t >
0, if s ∈ D ( X, C ), then exp (cid:0) − t (cid:3) X BC (cid:1) s ∈ Ω ( X, C ). If s ∈ ker ∂ ∩ ker ∂ , by (2.9), (2.12), we deduce that(2.13) exp (cid:0) − t (cid:3) X BC (cid:1) s − s = − ∂∂ (cid:90) t (cid:0) ∂∂ (cid:1) ∗ exp (cid:0) − u (cid:3) X BC (cid:1) sdu. Therefore if s ∈ D ( X, C ) lies in ker ∂ ∩ ker ∂ , and if { s } ∈ H BC ( X, C ) isthe corresponding Bott-Chern class, the form exp (cid:0) − t (cid:3) X BC (cid:1) s ∈ Ω ( X, C )is a smooth representative of { s } . Also, as t →
0, exp (cid:0) − t (cid:3) X BC (cid:1) s con-verges to s in D ( X, C ). Proposition 2.2.
If for n ∈ N , s n ∈ D ( X, C ) lies in ker ∂ ∩ ker ∂ ,if s ∈ D ( X, C ) and s n → s in D ( X, C ) , then s ∈ ker ∂ ∩ ker ∂ , and { s n } → { s } in H BC ( X, C ) .Proof. It is obvious that s ∈ ker ∂ ∩ ker ∂ . For t >
0, exp (cid:0) − t (cid:3) X BC (cid:1) s n converges to exp (cid:0) − t (cid:3) X BC (cid:1) s in Ω ( X, C ). Also the operator exp (cid:0) − t (cid:3) X BC (cid:1) does not change the Bott-Chern class. This completes the proof of ourproposition. (cid:3) Functorial properties of Bott-Chern cohomology.
Let Y be a compact complex manifold of dimension n (cid:48) , and let f : X → Y be a holomorphic map. Then f ∗ maps Ω ( Y, C ) into Ω ( X, C ) as amorphism of bigraded algebras. Therefore f ∗ induces a morphism ofbigraded algebras H BC ( Y, C ) → H BC ( X, C ). Also f ∗ preserves thecorresponding real vector spaces.By duality, f ∗ maps D • , • ( X, C ) into D n (cid:48) − n + • ,n (cid:48) − n + • ( Y, C ), and itpreserves the associated real vector spaces. Since the Bott-Chern coho-mology can be defined using currents, we get a morphism of bigradedvector spaces f ∗ : H • , • BC ( X, C ) → H n (cid:48) − n + • ,n (cid:48) − n + • BC ( Y, C ). This makessense, in spite of the fact that, in general, f ∗ does not act on D ( Y, C ),and f ∗ does not map Ω ( X, C ) into Ω ( Y, C ) .Let Y be the total space of T ∗ R Y , so that Y is viewed as the zerosection of Y . Let D f ( Y, C ) be the set of currents s on Y such that itswave-front set WF ( s ) ∈ Y \ Y does not intersect N f = f ∗ T R X ⊥ . ByH¨ormander [H83, Theorem 8.2.4], if s ∈ D f ( Y, C ) is a current on Y ,the current f ∗ s is well-defined. If s n | n ∈ N ∈ Ω ( Y, C ) is a microlocal ap-proximation of s in D f ( Y, C ) , then f ∗ s n converges to f ∗ s in D ( X, C ). Still, if f is a submersion, both assertions are true. This is the set of the ( f ( x ) , η ) | x ∈ X,η ∈ T ∗ R ,f ( x ) Y such that f ∗ ( x ) η = 0. This means that there is there is a closed conic subset Γ ⊂ Y such that Γ ∩ N f = ∅ , WF ( s ) ⊂ Γ , WF ( s n ) ⊂ Γ, and s n | n ∈ N converges to s in D Γ ( Y ) in the sense of[H83, Definition 8.2.2]. By [H83, Theorem 8.2.3], given s as before, such a sequence s n | n ∈ N always exists. OHERENT SHEAVES AND RRG 13 If s ∈ D • , • f ( Y, C ) is closed, the above shows that f ∗ s ∈ D • , • ( X, C ) isclosed, and that { f ∗ s } = f ∗ { s } in H • , • BC ( X, C ).In [H83, Theorem 8.2.10], the above argument is used to show thatif X is the total space of T ∗ R X , if s, s (cid:48) are two currents in D ( X, C )such that there are no ( x, ξ ) ∈ X such that ( x, ξ ) ∈ WF ( s ) , ( x, − ξ ) ∈ WF ( s (cid:48) ), the product of currents ss (cid:48) is well-defined and verifies suitablecontinuity properties. In particular, if s, s (cid:48) ∈ D • , • ( X, C ) are closed, ss (cid:48) is still closed and moreover,(2.14) { ss (cid:48) } = { s } { s (cid:48) } in H • , • BC ( X, C ) . Bott-Chern characteristic classes of holomorphic vectorbundles.
Here we follow [BGS88a, Section (e)]. Let (cid:0) F, ∇ F (cid:48)(cid:48) (cid:1) be aholomorphic vector bundle of rank m on X . Let g F be a Hermitianmetric on F , let ∇ F = ∇ F (cid:48)(cid:48) + ∇ F (cid:48) denote the corresponding Chernconnection on F . Then ∇ F (cid:48)(cid:48) , ∇ F (cid:48) act as odd operators on Ω ( X, F ).Let R F be its curvature. Then R F is a (1 , X with values inskew-adjoint sections of End ( F ). Then ∇ F (cid:48)(cid:48) , = 0 , ∇ F (cid:48) , = 0 , R F = (cid:2) ∇ F (cid:48)(cid:48) , ∇ F (cid:48) (cid:3) . (2.15)By (2.15), we obtain the Bianchi identities, (cid:2) ∇ F (cid:48)(cid:48) , R F (cid:3) = 0 , (cid:2) ∇ F (cid:48) , R F (cid:3) = 0 . (2.16)Let P ( B ) be an invariant polynomial on gl ( m, C ) under the adjointaction of GL ( m, C ). The polynomial P is said to be real if(2.17) P ( B ) = P (cid:0) B (cid:1) . Put(2.18) P (cid:0) F, g F (cid:1) = P (cid:0) − R F / iπ (cid:1) . Then the form P (cid:0) F, g F (cid:1) lies in Ω (=) ( X, C ). By Chern-Weil theory,the form P (cid:0) F, g F (cid:1) is closed, and its cohomology class does not dependon g F . This characteristic class will be denoted P ( F ) ∈ H even ( X, C ).If P is a real polynomial, then P (cid:0) F, g F (cid:1) ∈ Ω (=) ( X, R ), and P ( F ) ∈ H even ( X, R ).Let M F be the space of metrics on F , and let d M F be the de Rhamoperator on M F . Then θ = (cid:0) g F (cid:1) − d M F g F is a 1-form on M F withvalues in self-adjoint sections of End ( F ) with respect to the given g F . Recall that [ ] is our notation for the supercommutator.
BISMUT , SHU
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Note that d M F P (cid:0) F, g F (cid:1) is a 1-form on M F with values in Ω (=) ( X, C ).Let P (cid:48) be the derivative of P . Put(2.19) γ P (cid:0) F, g F (cid:1) = (cid:28) P (cid:48) (cid:18) − R F iπ (cid:19) , θ (cid:29) . Then γ P (cid:0) F, g F (cid:1) a 1-form on M F with values in Ω (=) ( X, C ). If P isreal, this form is also real.Let us recall a result of Bott-Chern [BoC65, Proposition 3.28], [BGS88a,Theorem 1.27]. Proposition 2.3.
The following identity holds: (2.20) d M F P (cid:0) F, g F (cid:1) = − ∂∂ iπ γ P (cid:0) F, g F (cid:1) . In particular the Bott-Chern class (cid:8) P (cid:0) F, g F (cid:1)(cid:9) ∈ H (=)BC ( X, C ) does notdepend on g F . If P is real, this class lies in H (=)BC ( X, R ) .Proof. Note that ∇ F (cid:48) depends on g F , and we have the identity(2.21) d M F ∇ F (cid:48) = − (cid:2) ∇ F (cid:48) , θ (cid:3) . Since R F = (cid:2) ∇ F (cid:48)(cid:48) , ∇ F (cid:48) (cid:3) , we get(2.22) d M F R F = − (cid:104) ∇ F (cid:48)(cid:48) , d M F ∇ F (cid:48) (cid:105) . Also(2.23) d M F P (cid:0) − R F (cid:1) = − (cid:68) P (cid:48) (cid:0) − R F (cid:1) , d M F R F (cid:69) . By (2.21)–(2.23), we obtain(2.24) d M F P (cid:0) − R F (cid:1) = − (cid:10) P (cid:48) (cid:0) − R F (cid:1) , (cid:2) ∇ F (cid:48)(cid:48) , (cid:2) ∇ F (cid:48) , θ (cid:3)(cid:3)(cid:11) . Using the Bianchi identities and (2.24), we get (2.20). If P is real, itis well known the form P (cid:0) F, g F (cid:1) is real.Let us now give another argument taken from [BGS88a, Theorem1.29] that shows that (cid:8) P (cid:0) F, g F (cid:1)(cid:9) does not depend on g F . On P ,we consider the canonical meromorphic coordinate z and the Poincar´e-Lelong equation(2.25) ∂ P ∂ P iπ log (cid:0) | z | (cid:1) = δ − δ ∞ . Let p , p be the projections from X × P on X, P . Let g F (cid:48) be anotherHermitian metric. Let g p ∗ F be a Hermitian metric on p ∗ F that restrictsto g F , g F (cid:48) on X × { } , X × {∞} . By (2.25), we get(2.26) ∂ X ∂ X iπ p ∗ (cid:2) p ∗ log (cid:0) | z | (cid:1) P (cid:0) p ∗ F, g p ∗ F (cid:1)(cid:3) = P (cid:0) F, g F (cid:1) − P (cid:0) F, g F (cid:48) (cid:1) , OHERENT SHEAVES AND RRG 15 which shows that (cid:8) P (cid:0) F, g F (cid:1)(cid:9) does not depend on g F . The proof ofour proposition is completed. (cid:3) Definition 2.4.
Let P BC ( F ) ∈ H (=)BC ( X, C ) be the Bott-Chern classof P (cid:0) F, g F (cid:1) .If P is real, then P BC ( F ) ∈ H (=)BC ( X, R ).Recall that if B ∈ gl ( m, C ), thenTd ( B ) = det (cid:20) B − e − B (cid:21) , (cid:98) A ( B ) = det (cid:20) B/ B/ (cid:21) , (2.27) ch ( B ) = Tr [exp ( B )] , c ( B ) = Tr [ B ] . The associated characteristic classes are called the Todd class, the (cid:98) A class, the Chern character, and the first Chern class. Also(2.28) Td ( B ) = (cid:98) A ( B ) e c ( B ) / . In Section 12, we will also use the real form of the class (cid:98) A . Moreprecisely, given k ∈ N , if so ( k ) is the Lie algebra of SO ( k ), if B ∈ so ( k ),set(2.29) (cid:98) A ( B ) = det (cid:20) B/ B/ (cid:21) / . To (cid:98) A ( B ), one can associate corresponding Pontryagin forms for realEuclidean vector bundles equipped with a metric connection.In the present paper, analogues of equation (2.20) will reappear inother contexts, that include vector bundles of infinite dimension. The derived category D bcoh ( X )The purpose of this Section is to recall basic properties of the derivedcategory D bcoh ( X ) of bounded complexes of O X -modules with coherentcohomology.This Section is organized as follows. In Subsection 3.1, we give thedefinition of D bcoh ( X ) and of the associated K -group K ( X ).In Subsection 3.2, if Y is a compact complex manifold, and if f : X → Y is a holomorphic map, we define the derived functor Lf ∗ :D bcoh ( Y ) → D bcoh ( X ).In Subsection 3.3, we consider the derived tensor products in D bcoh ( X ).Finally, in Subsection 3.4, we recall the properties of the derivedfunction Rf ∗ : D bcoh ( X ) → D bcoh ( Y ). The theory of
Bott-Chern classes developed in [BGS88a] is a secondary versionof Bott-Chern cohomology. It is an analogue of Chern-Simons theory for holomor-phic vector bundles. Here, this secondary theory will not play any role.
BISMUT , SHU
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Definition of the derived category D bcoh ( X ) . Let X be a com-pact complex manifold. Let coh ( X ) be the abelian category of O X -coherent sheaves on X , and let K (coh( X )) denote the correspondingGrothendieck group. In the sequel, we use the notation (3.1) K ( X ) = K (coh ( X )) . Let C b ( X ) be the category of bounded complexes of O X -modules.Let D b (X) be the corresponding derived category.Let C bcoh ( X ) be the full subcategory of C b ( X ) whose objects havecoherent cohomology. Let D bcoh ( X ) be the corresponding derived cat-egory. Then D bcoh ( X ) is the full subcategory of D b ( X ) whose objectshave coherent cohomology.The same arguments as in [St20, Tag 0FCP] show that the map(3.2) E ∈ D bcoh ( X ) → (cid:88) i ( − i H i E ∈ K ( X )induces an isomorphism of K -groups,(3.3) K (D bcoh ( X )) (cid:39) K ( X ) . Pull-backs.
Let Y be another compact complex manifold, andlet f : X → Y be a holomorphic map.By [St20, Tag 0095], if E ∈ C bcoh ( Y ), one can define its pull-back f ∗ E , which is a complex of O X -modules, by the formula(3.4) f ∗ E = f − E ⊗ f − O Y O X . By Grauert-Remmert [GrR84, Section 1.2.6], f ∗ E ∈ C bcoh ( X ). We candefine the left-derived functor Lf ∗ , which to E ∈ D bcoh ( Y ) associates Lf ∗ E ∈ D bcoh ( X ).If E is a bounded complex of flat O Y -modules, by [St20, Tag 06YJ],we have(3.5) Lf ∗ E = f ∗ E . Also Lf ∗ induces a morphism on Grothendieck groups,(3.6) f ! : K ( Y ) → K ( X ) . If Z is another compact complex manifold, if g : Y → Z is a holo-morphic map, using [Bor87, Proposition I.9.15], there is a canonicalisomorphism between the functors Lg ∗ Lf ∗ and L ( f g ) ∗ . In particular,we get an identity of morphisms of Grothendieck groups,(3.7) f ! g ! = ( gf ) ! : K ( Z ) → K ( X ) . In the introduction, we used the notation K · ( X ). OHERENT SHEAVES AND RRG 17
Tensor products.
Let E , F be objects in D bcoh ( X ). In [St20, Tag064M], a derived tensor product E (cid:98) ⊗ L O X F , also an object in D bcoh ( X )is defined. By [St20, Tag 079U], if Y is a compact complex manifold,and if f : X → Y is holomorphic, if E , F are objects in D bcoh ( Y ), then(3.8) Lf ∗ (cid:16) E (cid:98) ⊗ L O Y F (cid:17) = Lf ∗ E (cid:98) ⊗ L O X Lf ∗ F . Let i : X → X × X be the diagonal embedding, and let p , p : X × X → X be the two projections. Since p i , p i are the identity in X , using the results of Subsection 3.2 and equation (3.8), we find thatthere is a canonical isomorphism,(3.9) E (cid:98) ⊗ L O X F (cid:39) Li ∗ (cid:16) Lp ∗ E (cid:98) ⊗ L O X × X Lp ∗ F (cid:17) . By [St20, Tag 06XY], if E , F are objects in D bcoh ( X ), if one of themconsists of flat modules over O X , then(3.10) E (cid:98) ⊗ L O X F = E (cid:98) ⊗ O X F . Direct images.
Let Rf ∗ be the right derived functor of the directimage f ∗ . By a theorem of Grauert [GrR84, Theorem 10.4.6], if E is anobject in D bcoh ( X ), Rf ∗ E is an object in D bcoh ( Y ).By [Bor87, Theorem 9.5] and by [D09, Proposition IV.4.14], if E isa bounded complex of soft O X -modules, then(3.11) Rf ∗ E = f ∗ E . Also Rf ∗ induces a morphism of Grothendieck groups,(3.12) f ! : K ( X ) → K ( Y ) . If Z is a compact complex manifold, and if g : Y → Z is a holo-morphic map, by [Bor87, Proposition I.9.15], there is a canonical iso-morphism between the functors Rg ∗ Rf ∗ and R ( gf ) ∗ . In particular, wehave an identity of morphisms of Grothendieck groups,(3.13) g ! f ! = ( gf ) ! : K ( X ) → K ( Z ) . Preliminaries on linear algebra and differentialgeometry
The purpose of this Section is to explain elementary facts of linearalgebra and differential geometry that will be used in the rest of thepaper. We have replaced the usual notation ⊗ L by (cid:98) ⊗ L , to underline the fact that theobjects in D bcoh ( X ) are Z -graded. BISMUT , SHU
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WEI
This Section is organized as follows. In Subsection 4.1, if W is acomplex vector space, we introduce the category C W of free Λ (cid:16) W ∗ (cid:17) -modules.In Subsection 4.2, we define the supertraces, and give its main prop-erties.In Subsection 4.3, we recall the construction of cones in homologicalalgebra.In Subsection 4.4, we construct generalized Hermitian metrics onobjects in C W .In Subsection 4.5, we recall elementary properties of the Cliffordalgebras of complex Hermitian vector spaces.Finally, in Subsection 4.6, we explain various relations of the deRham operator to connections with nonzero torsion on the tangentbundle of a real manifold.4.1. Filtered vector space and exterior algebras.
Let W be afinite-dimensional complex vector space of dimension m , and let W R be the corresponding real vector space. If W C = W R ⊗ R C is itscomplexification, then W C = W ⊕ W .If r, r (cid:48) ∈ Z , r ≤ r (cid:48) , let E = (cid:76) r (cid:48) i = r E i be a Z -graded vector space onwhich Λ p (cid:16) W ∗ (cid:17) acts on the left as linear morphisms of degree p (i.e.,they increase the degree by p ). We can also define a right action ofΛ (cid:16) W ∗ (cid:17) on E , so that if e ∈ E, α ∈ Λ (cid:16) W ∗ (cid:17) , then(4.1) eα = ( − deg e deg α αe. For 0 ≤ p ≤ m , put(4.2) F p E = Λ p (cid:16) W ∗ (cid:17) E. The F p E form a decreasing filtration(4.3) F E = E ⊃ F E ⊃ . . . ⊃ F m +1 E = 0 . Put(4.4) E p,q = F p E p + q /F p +1 E p + q . For 0 ≤ p (cid:48) ≤ m , Λ p (cid:48) (cid:16) W ∗ (cid:17) maps E p,q into E p + p (cid:48) ,q surjectively.For r ≤ q ≤ r (cid:48) , set(4.5) D q = E ,q = E q /F E q . Then Λ p ( W ∗ ) maps D q into E p,q surjectively. OHERENT SHEAVES AND RRG 19
In the sequel, we assume that we have the identification of leftΛ (cid:16) W ∗ (cid:17) -modules(4.6) E • , • = Λ • (cid:16) W ∗ (cid:17) (cid:98) ⊗ D • . We will call D the diagonal vector space associated with E . There is acanonical degree preserving morphism E → D , and we have the exactsequence(4.7) 0 → F E → E → D → . Note that a basis of D lifts in E to a basis of E as a free Λ (cid:16) W ∗ (cid:17) -module.For r ≤ i ≤ r (cid:48) , put E i = (cid:77) p + q = i E p,q , E = r (cid:48) (cid:77) i = r E i . (4.8)Then E is a Z -graded filtered vector space, which has the same prop-erties as E , with the same D .Let E be another Z -graded vector space like E . We underline theobjects associated with E . If k ∈ Z , if φ is a linear morphism from E into E that maps E • into E • + k , we will say that φ is of degree k .Let Hom ( E, E ) be the set of linear maps from E into E such that if α ∈ Λ (cid:16) W ∗ (cid:17) , then(4.9) φα − ( − deg α deg φ αφ = 0 , If α ∈ Λ (cid:16) W ∗ (cid:17) , e ∈ E , equation (4.9) is equivalent to the fact that(4.10) φ ( eα ) = ( φe ) α. Then Hom (
E, E ) is a Z -graded algebra. There is a left action ofΛ p (cid:16) W ∗ (cid:17) on Hom ( E, E ) by morphisms of degree p .Given W , let C W be the category consisting of the above objects E that verify (4.6), and of the morphisms in Hom ( E, E ). If
E, E are objects in C W , then Hom ( E, E ) is still an object in C W , and thecorresponding diagonal vector space is given by Hom ( D, D ).If E = Λ (cid:16) W ∗ (cid:17) , then D = C , and Hom (cid:16) Λ (cid:16) W ∗ (cid:17) , E (cid:17) = E . Put(4.11) E † = Hom (cid:16) E, Λ (cid:16) W ∗ (cid:17)(cid:17) . BISMUT , SHU
SHEN , AND ZHAOTING
WEI
Then D † = D ∗ . If e ∈ E , the map φ ∈ Hom (cid:16) E, Λ (cid:16) W ∗ (cid:17)(cid:17) → ( − deg φ deg e φe ∈ Λ (cid:16) W ∗ (cid:17) gives the identification(4.12) (cid:0) E † (cid:1) † = E. Also(4.13) E † = Λ (cid:16) W ∗ (cid:17) (cid:98) ⊗ D ∗ . If W (cid:48) has the same properties as W , by [Ke06, Section 2.3], there isa tensor product that is a functor from C W (cid:98) ⊗ C W (cid:48) to C W ⊕ W (cid:48) . Proposition 4.1. If E, E are objects in C W , the morphism φ ∈ Hom (
E, E ) is an isomorphism if and only if φ induces an isomorphism from D into D .Proof. We only need to show that if φ maps isomorphically D into D , then it acts isomorphically from E into E . By (4.4), (4.6), φ alsoacts as an isomorphism from F p E/F p +1 E into F p E/F p +1 E . A recur-sion argument shows that φ is an isomorphism from F p E/F p + i E into F p E/F p + i E . For p = 0 , i = m + 1, we obtain our proposition. (cid:3) Observe that Aut ( E ) is a Lie group, and its Lie algebra is justEnd ( E ). There is a surjective group homomorphism ρ : Aut ( E ) → Aut ( D ), that induces a corresponding morphism of Lie algebras End ( E ) → End ( D ). PutAut ( E ) = { g ∈ Aut ( E ) , ρg = 1 } , End ( E ) = { f ∈ End ( E ) , ρf = 0 } . (4.14)Then Aut ( E ) is a Lie subgroup of Aut ( E ) with Lie algebra End ( E ).Let Aut ( E ) be the Lie subgroup of the elements of Aut ( E ) of degree0. Its Lie algebra End ( E ) consists of the elements of End ( E ) of degree0. We use the notationAut ( E ) = Aut ( E ) ∩ Aut ( E ) , End ( E ) = End ( E ) ∩ End ( E ) . (4.15)If g ∈ Aut ( E ), then ( g − m +1 = 1. Similarly if f ∈ End ( E ),then f m +1 = 0. Proposition 4.2. If g ∈ Aut ( E ) , there exists a unique pair h ∈ Aut ( D ) , f ∈ End ( E ) such that (4.16) g = exp ( f ) h, and h = ρg . OHERENT SHEAVES AND RRG 21
Proof.
Replacing g by g ( ρg ) − , we may as well assume that g ∈ Aut ( E ).Then g − f = log g = log (1 + ( g − f ∈ End ( E ). The proof of our proposition is completed. (cid:3) Observe that(4.17) End ( E ) = Λ (cid:16) W ∗ (cid:17) (cid:98) ⊗ End ( D ) . Any f ∈ End ( E ) can be written in the form f = m (cid:88) f i , f i ∈ Λ i (cid:16) W ∗ (cid:17) (cid:98) ⊗ End ( D ) . (4.18)Then f ∈ Aut ( E ) if and only if f ∈ Aut ( D ).By (4.7), there are non-canonical splittings,(4.19) E q = D q ⊕ F E q . By (4.6), (4.19), we get the non-canonical identification of Z -gradedΛ (cid:16) W ∗ (cid:17) -modules,(4.20) E (cid:39) E . Let Hom ( D, F E ) be the homomorphisms of degree 0 of D in F E . The other splittings are obtained via the action of A ∈ Hom ( D, F E ).If we consider another splitting as in (4.19) associated with A ∈ Hom ( D, F E ), the corresponding E is unchanged, but the identifi-cation to E is different. This way, we obtain an automorphism of E , g = 1 + A ∈ Aut ( E ).4.2. Supercommutators and supertraces.
Let A = A + ⊕ A − be a Z -graded algebra. If a, b ∈ A , recall that [ a, b ] is the supercommutatorof a, b .If E is an object in C W , then End ( E ) = Hom ( E, E ) is a Z -gradedalgebra, and so it inherits a corresponding Z -grading. If we fix split-tings as in (4.20), then End ( E ) = End ( E ), and End ( E ) is given by(4.17).Let τ = ± D that defines its Z -grading. If A ∈ End ( D ), its supertrace Tr s [ A ] ∈ C is defined by(4.21) Tr s [ A ] = Tr [ τ A ] . We extend Tr s to a map from Λ ( W ∗ C ) (cid:98) ⊗ End ( D ) into Λ ( W ∗ C ), with theconvention that if α ∈ Λ ( W ∗ C ) , A ∈ End ( D ), then(4.22) Tr s [ αA ] = α Tr s [ A ] . By [Qu85], Tr s vanishes on supercommutators in Λ ( W ∗ C ) (cid:98) ⊗ End ( D ). BISMUT , SHU
SHEN , AND ZHAOTING
WEI
Proposition 4.3.
The linear map Tr s induces a morphism Λ ( W ∗ ) (cid:98) ⊗ End ( E ) → Λ ( W ∗ C ) that does not depend on the splitting (4.19), and vanishes onsupercommutators.Proof. As we saw at the end of Subsection 4.1, the choice of anothersplitting is equivalent to the choice of g ∈ Aut ( E ). Such a g isnecessarily even, so that if a ∈ Λ ( W ∗ C ) (cid:98) ⊗ End ( D ), then(4.23) Tr s (cid:2) gag − (cid:3) = Tr s [ a ] , from which our proposition follows. (cid:3) Morphisms and cones. If (cid:0) Q, d Q (cid:1) is a bounded complex of com-plex vector spaces with differential d Q of degree 1, we denote by HQ its cohomology.Let (cid:0) Q, d Q (cid:1) , (cid:0) Q, d Q (cid:1) be two such bounded complexes of complexvector spaces. Let φ : Q → Q be a morphism of complexes, so that φ preserves the degree, so that(4.24) φd Q = d Q φ. Then φ induces a morphism φ : HQ → HQ . Also φ is said to be aquasi-isomorphism if φ : HQ → HQ is an isomorphism.Put(4.25) C = cone (cid:0) Q, Q (cid:1) . Then (cid:0)
C, d C (cid:1) is also a bounded complex such that(4.26) C • = Q • +1 ⊕ Q • . The morphism φ can be identified with an endomorphism of C , thatmaps C • into C • +1 , i.e., φ can be viewed as a morphism of degree 1 inEnd ( C ), and the differential d Cφ is given by(4.27) d Cφ = (cid:20) d Q φ ( − deg d Q (cid:21) . We have the exact sequence of complexes,(4.28) 0 (cid:47) (cid:47) Q • (cid:47) (cid:47) C • (cid:47) (cid:47) Q • +1 (cid:47) (cid:47) . There is a corresponding long exact sequence in cohomology,(4.29) H • Q (cid:47) (cid:47) H • C (cid:47) (cid:47) H • +1 Q φ ( − • +1 (cid:47) (cid:47) H • +1 Q .
By (4.29), φ is a quasi-isomorphism if and only if HC = 0. OHERENT SHEAVES AND RRG 23
For t ∈ C , let M t ∈ End ( C ) be given in matrix form by(4.30) M t = (cid:20) t (cid:21) . For t (cid:54) = 0, M t is invertible, and(4.31) d Ctφ = M t d Cφ M − t , so that M t is an isomorphism from (cid:0) C, d Cφ (cid:1) into (cid:0) C, d
Ctφ (cid:1) .4.4.
Generalized Hermitian metrics.
We make the same assump-tions as in Subsection 4.1. If e ∈ W ∗ C , set(4.32) (cid:101) e = − e. We still denote by (cid:101) the corresponding anti-automorphism of algebrasof Λ ( W ∗ C ), so that if α ∈ Λ p ( W ∗ C ), then(4.33) (cid:101) α = ( − p ( p +1) / α. Put(4.34) α ∗ = (cid:101) α. If A ∈ End ( D ), let (cid:101) A ∈ End ( D ∗ ) be its transpose. Then (cid:101) ex-tends to an anti-automorphism from End ( E ) = Λ (cid:16) W ∗ (cid:17) (cid:98) ⊗ End ( D ) toEnd (cid:16) E † (cid:17) = Λ (cid:16) W ∗ (cid:17) (cid:98) ⊗ End ( D ∗ ).If f ∈ End ( D ), we define f ∗ ∈ End (cid:16) D ∗ (cid:17) by the formula(4.35) f ∗ = (cid:101) f . Our conventions here coincide with the ones in [BL95, Section 1 (c)]and [B13, Section 3.5].Recall that(4.36) Λ ( W ∗ C ) = Λ ( W ∗ ) (cid:98) ⊗ Λ (cid:0) W ∗ (cid:1) , and that the natural grading Λ ( W ∗ C ) is given by the sum of the degreesin Λ ( W ∗ ) and Λ (cid:0) W ∗ (cid:1) . If α ∈ Λ i ( W ∗ C ), we write(4.37) deg α = i. We will often count the degree in Λ p (cid:16) W ∗ (cid:17) , Λ q ( W ∗ ) to be p, − q ,and we introduce the corresponding degree deg − on Λ ( W ∗ C ) so that if α ∈ Λ p (cid:16) W ∗ (cid:17) , β ∈ Λ q ( W ∗ ),(4.38) deg − α ∧ β = p − q. BISMUT , SHU
SHEN , AND ZHAOTING
WEI
Most of the time, we will use this second grading on Λ ( W ∗ C ). We willstill use the notation Λ i ( W ∗ C ) for the forms of total degree equal to i .In the sequel, elements in D i , D i ∗ will be said to be of degree i . If A ∈ Hom (cid:16)
D, D ∗ (cid:17) = D ∗ ⊗ D ∗ , we count its degree as the difference of thedegrees in D ∗ and D ∗ . Let A ∗ ∈ Hom (cid:16)
D, D ∗ (cid:17) denote the conjugate ofthe transpose of A . We equip Λ ( W ∗ C ) (cid:98) ⊗ Hom (cid:16)
D, D ∗ (cid:17) with the obviousantilinear involution ∗ and with the degree induced by deg − on Λ ( W ∗ C )and by the above degree on Hom (cid:16) D, D ∗ (cid:17) . Under ∗ , this degree ischanged into its negative. An element of Λ ( W ∗ C ) (cid:98) ⊗ Hom (cid:16)
D, D ∗ (cid:17) issaid to be self-adjoint if it is invariant under ∗ .If h ∈ Λ ( W ∗ C ) (cid:98) ⊗ Hom (cid:16)
D, D ∗ (cid:17) , we can write h in the form h = m (cid:88) i =0 h i , h i ∈ Λ i ( W ∗ C ) (cid:98) ⊗ Hom (cid:16)
D, D ∗ (cid:17) . (4.39) Definition 4.4.
An element h ∈ Λ ( W ∗ C ) (cid:98) ⊗ Hom (cid:16)
D, D ∗ (cid:17) is said to bea generalized metric on D if it is of degree 0, self-adjoint, and such that h defines a Z -graded Hermitian metric on the Z -graded vector space D . Let M D be the set of generalized metrics on D .A metric h ∈ M D is said to be pure if h = h .If h ∈ M D , then h − ∈ M D ∗ , h − ∈ M D ∗ . If f ∈ Λ ( W ∗ C ) (cid:98) ⊗ End ( D ),we define f ∗ ∈ Λ ( W ∗ C ) (cid:98) ⊗ End (cid:16) D ∗ (cid:17) by the same procedure as before.Observe that Aut ( E ) acts on M D so that if g ∈ Aut ( E ), theaction of g − on M D is given by(4.40) g † h = g ∗ hg. Clifford algebras.
Let V be a finite-dimensional complex vectorspace of dimension n , and let V R be its real form. Let V C = V R ⊗ R C be the complexification of V R , so that V C = V ⊕ V .Let g V ∗ be a Hermitian metric on V ∗ , and let g V ∗ R be the correspond-ing scalar product on V ∗ R . Then g V ∗ identifies V ∗ and V . Let c ( V ∗ R )be the Clifford algebra of (cid:0) V ∗ R , g V ∗ R (cid:1) , which is generated by 1 , f ∈ V ∗ R , This just means that the D i are mutually orthogonal. For convenience, we prefer to introduce the Clifford algebra of V ∗ R instead ofthe usual c ( V R ). OHERENT SHEAVES AND RRG 25 and the commutation relations for f, f (cid:48) ∈ V ∗ R , (4.41) f f (cid:48) + f (cid:48) f = − (cid:104) f, f (cid:48) (cid:105) g V ∗ R . As a Z -graded vector space, c ( V ∗ R ) is canonically isomorphic to theexterior algebra Λ ( V ∗ R ). Let f , . . . , f n be an orthonormal basis of V ∗ R .If 0 ≤ p ≤ n , and if 1 ≤ i < . . . < i p ≤ n , the identification maps f i . . . f i p ∈ c ( V ∗ R ) to f i ∧ . . . ∧ f i p ∈ Λ ( V ∗ R ). If B ∈ Λ ( V ∗ R ), we denoteby c B the corresponding element in c ( V ∗ R ).If f ∈ V ∗ , if f ∗ = g V ∗ f ∈ V , put c (cid:0) f (cid:1) = f ∧ , c ( f ) = − i f ∗ . (4.42)By (4.42), Λ (cid:16) V ∗ (cid:17) is a c ( V ∗ R )-Clifford module. We have the identifica-tion of Z -graded algebras,(4.43) c ( V ∗ R ) ⊗ R C = End (cid:16) Λ (cid:16) V ∗ (cid:17)(cid:17) . In the sequel, we will not distinguish between the two sides of (4.43). Inparticular, if B ∈ Λ ( V ∗ C ), c B is viewed as an element of End (cid:16) Λ (cid:16) V ∗ (cid:17)(cid:17) .In particular, if B ∈ Λ (cid:16) V ∗ (cid:17) , c B is just the exterior product by B .Let U be another finite-dimensional complex vector space. We willview Λ ( U ∗ R ) as an algebra acting upon itself by left multiplication.Then(4.44) Λ ( U ∗ R ⊕ V ∗ R ) = Λ ( U ∗ R ) (cid:98) ⊗ Λ ( V ∗ R ) . Also Λ ( U ∗ R ⊕ V ∗ R ) can be identified with Λ ( U ∗ R ) (cid:98) ⊗ c ( V ∗ R ). If B ∈ Λ ( U ∗ C ⊕ V ∗ C ),let c B the associated element in Λ ( U ∗ C ) (cid:98) ⊗ End (cid:16) Λ (cid:16) V ∗ (cid:17)(cid:17) . If B ∈ Λ (cid:16) U ∗ ⊕ V ∗ (cid:17) , c B is still multiplication by B .4.6. A simple remark of differential geometry.
Let Z be a realmanifold, with real tangent bundle T Z , and let ∇ T Z be a connectionon
T Z , with torsion T . Let ∇ Λ( T ∗ Z ) denote the induced connection onΛ ( T ∗ Z ). Let ∇ Λ( T ∗ Z )a denote the antisymmetrization of ∇ Λ( T ∗ Z ) , thatacts on Ω ( Z, R ) and increases the degree by 1.Let d Z be the de Rham operator on Z . Then i T is a 2-form on Z with values in contraction operators . Let ( f α ) be a basis of T Z , and The commutation relations are usually written with an extra factor 2 in theright-hand side. In complex geometry, it is more natural to drop the factor 2. We will always use the normal ordering of such operators, so that contractionoperators act before multiplication by forms.
BISMUT , SHU
SHEN , AND ZHAOTING
WEI let ( f α ) be the corresponding dual basis of T ∗ Z . Then i T is given by(4.45) i T = 12 f α f β i T ( f α ,f β ) . We have the classical identity(4.46) d Z = ∇ Λ( T ∗ Z )a + i T . For a ∈ R , let ∇ T Z,a be the connection on
T Z such that if
U, V ∈ T Z ,(4.47) ∇ T Z,aU V = ∇ T ZU V + aT ( U, V ) . Then the torsion of ∇ T Z,a is given by (1 + 2 a ) T . Let ∇ Λ( T ∗ Z ) ,a be theconnection on Λ ( T ∗ Z ) that is induced by ∇ T Z,a .In the sequel, we define the operator i T ( U, · ) using normal ordering,i.e.,(4.48) i T ( U, · ) = f α i T ( U,f α ) . By (4.47), if U ∈ T Z ,(4.49) ∇ Λ( T ∗ Z ) ,aU = ∇ Λ( T ∗ Z ) U − ai T ( U, · ) . We have the identity of connections on Λ ( T ∗ Z ),(4.50) ∇ Λ( T ∗ Z ) ,a = ∇ Λ( T ∗ Z ) − a (cid:104) T ( · , f β ) , f γ (cid:105) f β i f γ . By (4.45), (4.50), we deduce that(4.51) ∇ Λ( T ∗ Z ) ,a a = ∇ Λ( T ∗ Z )a − ai T . By (4.46), (4.51), we get(4.52) d Z = ∇ Λ( T ∗ Z ) , − / . If U is a smooth section of T Z , then(4.53) [
U, V ] = ∇ T ZU V − T ( U, V ) − ∇ T ZV U. Equation (4.53) can be rewritten in the form(4.54) [
U, V ] = ∇ T Z, − U V − ∇ T ZV U. We define the operator i ∇ TZ U using normal ordering, i.e.,(4.55) i ∇ TZ U = f α i ∇ TZfα U . By (4.54), when acting on Ω ( Z, R ), the Lie derivative operator L U is given by(4.56) L U = i ∇ TZ U + ∇ Λ( T ∗ Z ) , − U . Combining (4.49) and (4.56), we obtain(4.57) L U = i ∇ TZ U + ∇ Λ( T ∗ Z ) U + i T ( U, · ) . OHERENT SHEAVES AND RRG 27
Equation (4.57) is compatible with equation (4.46) and with Cartanformula L U = (cid:2) d Z , i U (cid:3) , which can also be written in the form(4.58) L U = (cid:2) ∇ Λ( T ∗ Z )a + i T , i U (cid:3) = (cid:2) ∇ Λ( T ∗ Z ) , − / , i U (cid:3) . Let Z be a complex manifold, let T Z be its holomorphic tangentbundle, and let T R Z be its real tangent bundle. Let g T Z be a Hermit-ian metric on
T Z , let ∇ T Z be the associated holomorphic Hermitianconnection on
T Z , and let ∇ T R Z be the induced connection on T R Z ,with torsion T . Then T splits as (2 , T Z and a(0 , T Z . For a ∈ R , ∇ T R Z,a preserves
T Z and
T Z . More precisely, if u, v ∈ T Z are smooth sections of
T Z , ∇ T Z,a (cid:48)(cid:48) = ∇ T Z (cid:48)(cid:48) , ∇ T Z,a (cid:48) u v = ∇ T Zu v + aT ( u, v ) . (4.59)Also equation (4.52) splits into separate formulas for ∂ Z , ∂ Z .5. The antiholomorphic superconnections of Block
The purpose of this Section is to describe the antiholomorphic su-perconnections introduced by Block [Bl10].This Section is organized as follows. In Subsection 5.1, we definethe antiholomorphic superconnections (cid:0)
E, A E (cid:48)(cid:48) (cid:1) on a compact complexmanifold X .In Subsection 5.2, we establish the local conjugation result of Block,which asserts that locally, A E (cid:48)(cid:48) can be reduced to a canonical form.This result is an extension of the Newlander-Nirenberg theorem forholomorphic vector bundles.In Subsection 5.3, when viewing E = (cid:0) E, A E (cid:48)(cid:48) (cid:1) as a sheaf of O X -complexes, we show that its cohomology is a Z -graded coherent sheaf,and that the natural spectral sequence degenerates at E .In Subsection 5.4, we show that the complex line det D has a naturalholomorphic structure, and that det D coincides with the determinantline of Knudsen-Mumford [KnM76].In Subsection 5.5, we show that H E = 0 if and only if for any x ∈ X , the complex ( D, v ) x is exact.In Subsection 5.6, given a morphism of antiholomorphic supercon-nections, we construct the corresponding cone.In Subsection 5.7, we construct the pull-backs of antiholomorphicsuperconnections.Finally, in Subsection 5.8, we obtain a tensor product of antiholo-morphic superconnections. BISMUT , SHU
SHEN , AND ZHAOTING
WEI
The antiholomorphic superconnections.
Let X be a compactcomplex manifold of dimension n . If r, r (cid:48) ∈ Z , r ≤ r (cid:48) , let E = (cid:76) r (cid:48) i = r E i be a finite-dimensional complex Z -graded vector bundle on X , whichis also a left Λ (cid:0) T ∗ X (cid:1) -module, so that elements of Λ p (cid:0) T ∗ X (cid:1) act on E as morphisms of degree p .We use the notation of Section 4, with W replaced by T X . Weassume that the D q as defined (4.5) have constant rank, so that thecorresponding diagonal D is a complex Z -graded vector bundle on X .Also we assume that the analogue of (4.6) holds.Then C ∞ ( X, E ) is a Z -graded vector space. The degree of an oper-ator acting on C ∞ ( X, E ) is counted as in Section 4.Let P denote the projection E → D . Then P preserves the Z -grading. Also P vanishes on F E .As in (4.8), set E i = (cid:77) p + q = i Λ p (cid:0) T ∗ X (cid:1) ⊗ D q , E = r (cid:48) (cid:77) i = r E i . (5.1)Then E is a Z -graded filtered vector bundle with the same D as E .We can write the analogue of (4.19) as a non-canonical splitting ofsmooth vector bundles,(5.2) E q = D q ⊕ F E q . As in (4.20), we get the smooth non-canonical identification of Z -gradedΛ (cid:0) T ∗ X (cid:1) -modules,(5.3) E (cid:39) E . Any other splitting is obtained from smooth sections of Hom ( D, F E ).Now we follow Quillen [Qu85, Section 2] and Block [Bl10, Definition2.4]. Definition 5.1.
A differential operator A E (cid:48)(cid:48) acting on C ∞ ( X, E ) issaid to be an antiholomorphic flat superconnection if it acts as adifferential operator of degree 1 on C ∞ ( X, E ) such that • If α ∈ Ω , • ( X, C ) , s ∈ C ∞ ( X, E ), then(5.4) A E (cid:48)(cid:48) ( αs ) = (cid:16) ∂ X α (cid:17) s + ( − deg α αA E (cid:48)(cid:48) s. • The following identity holds:(5.5) A E (cid:48)(cid:48) , = 0 . In [Bl10, Sections 2 and 4], Block writes instead that A E (cid:48)(cid:48) is a Z -connection. OHERENT SHEAVES AND RRG 29
In the sequel, we will omit ‘flat’, and say instead that A E (cid:48)(cid:48) is anantiholomorphic superconnection.The operator A E (cid:48)(cid:48) preserves the filtration F . In particular A E (cid:48)(cid:48) actson D = E/F E like a smooth section v ∈ End ( D ) which is of de-gree 1, and such that v = 0. Then P is a morphism of complexes (cid:0) C ∞ ( X, E ) , A E (cid:48)(cid:48) (cid:1) → ( C ∞ ( X, D ) , v ).If g is a smooth section of Aut ( E ), then A E (cid:48)(cid:48) = gA E (cid:48)(cid:48) g − is anotherantiholomorphic superconnection on E . In the sequel, we will writethat A E (cid:48)(cid:48) (cid:39) A E (cid:48)(cid:48) .Assume for the moment that we fix a non-canonical identification asin (5.2), (5.3). Let A E (cid:48)(cid:48) be the corresponding antiholomorphic super-connection on E . We can write A E (cid:48)(cid:48) in the form(5.6) A E (cid:48)(cid:48) = v + ∇ D (cid:48)(cid:48) + (cid:88) i ≥ v i , where ∇ D (cid:48)(cid:48) a degree preserving antiholomorphic connection on D ,and for i = 0 or i ≥ v i ∈ Λ i (cid:0) T ∗ X (cid:1) (cid:98) ⊗ End − i ( D ). Since A E (cid:48)(cid:48) , = 0,from (5.6), we get v = 0 , (cid:2) ∇ D (cid:48)(cid:48) , v (cid:3) = 0 , ∇ D (cid:48)(cid:48) , + [ v , v ] = 0 . (5.7)Put(5.8) B = v + (cid:88) i ≥ v i . Then B is a section of degree 1 of Λ (cid:0) T ∗ X (cid:1) (cid:98) ⊗ End ( D ), and (5.6) canbe written in the form,(5.9) A E (cid:48)(cid:48) = ∇ D (cid:48)(cid:48) + B. Let g be a smooth section of Aut ( E ) associated with a differentsplitting in (5.2), (5.3). Let A E (cid:48)(cid:48) be the superconnection on E thatcorresponds to A E (cid:48)(cid:48) as before. Then(5.10) A E (cid:48)(cid:48) = gA E (cid:48)(cid:48) g − . Then A E (cid:48)(cid:48) has an expansion similar to (5.6), in which the correspondingterms will also be underlined. Note that v = v , and also that thereis a smooth section α of T ∗ X (cid:98) ⊗ End − ( D ) such that(5.11) ∇ D (cid:48)(cid:48) = ∇ D (cid:48)(cid:48) + [ α, v ] . In general, its square does not vanish, i.e., it does not define a holomorphicstructure on D . The distinction between antiholomorphic connection and holomor-phic structure will reappear in the sequel. BISMUT , SHU
SHEN , AND ZHAOTING
WEI
Example 5.2.
Here are two trivial cases of antiholomorphic supercon-nections: C X = (cid:16) Λ (cid:0) T ∗ X (cid:1) , ∂ X (cid:17) , C X = (cid:16) Λ ( T ∗ C X ) , ∂ X (cid:17) . (5.12)In the first case, D = C , in the second case, D = Λ ( T ∗ X ). In bothcases, we have canonical identifications E = E .5.2. A conjugation result.
The following extension of the Newlander-Nirenberg theorem has been established in [Bl10, p.17].
Theorem 5.3. If (cid:0) E, A E (cid:48)(cid:48) (cid:1) is an antiholomorphic superconnection, if x ∈ X , there is an open neighborhood U of x , a holomorphic structure ∇ D | U (cid:48)(cid:48) on D | U such that ∇ D | U (cid:48)(cid:48) v = 0 , and moreover, (5.13) ( E, A E (cid:48)(cid:48) ) | U (cid:39) (cid:16) Λ( T ∗ U ) ⊗ D | U , v + ∇ D | U (cid:48)(cid:48) (cid:17) . Proof.
We will proceed as in proof of the Newlander-Nirenberg theo-rem [Do88, Subsection 2.2.2]. Recall that dim X = n . Let ( z , . . . , z n )be a coordinate system near x such that x is represented by z = 0. For r > U r = { z ∈ C n : | z i | < r } be the correspondingpolydisc. For r small enough, we can view U r as an open neighbor-hood of x in X , such that D | U r is a trivial smooth vector bundle. Inparticular, D is equipped with the holomorphic structure ∂ . By (5.6),on U r , there is a smooth section B of Λ (cid:0) C n (cid:1) (cid:98) ⊗ End ( D ) of degree 1such that(5.14) A E (cid:48)(cid:48) | U r = ∂ + B . If B ( ≥ is the sum of the components of B of degree ≥ (cid:0) C n (cid:1) ,then(5.15) B = v + B ( ≥ . For t ∈ C , | t | ≤
1, consider the dilation z → δ t ( z ) = tz . Replacing r by tr , and A E (cid:48)(cid:48) | U r by its conjugate by δ ∗ t , we may as well assume that (cid:13)(cid:13)(cid:13) B ( ≥ (cid:13)(cid:13)(cid:13) C b ( U r ) is arbitrary small.We will show by induction that for 1 ≤ i ≤ n , there are smoothsections J i of Aut (Λ (cid:0) C n (cid:1) (cid:98) ⊗ D ), and antiholomorphic superconnections ∂ + B i on U r such that ( J i , B i ) do not contain the Grassmannian variable dz j , j ≤ i , and that(5.16) J − i (cid:0) ∂ + B i − (cid:1) J i = ∂ + B i . If J = J · · · J n , we will obtain part of our theorem by conjugating A E (cid:48)(cid:48) by J − . OHERENT SHEAVES AND RRG 31
For i = 1, we need to find J such that J and ∂J + B J do notcontain dz , in which case B = J − (cid:0) ∂J (cid:1) + J − B J . (5.17)In the sequel we view Λ (cid:16) C n − (cid:17) as being generated by dz , . . . , dz n . If z = ( z , z (cid:48) ) ∈ U r , with z (cid:48) ∈ C n − fixed for the moment, we have to findan invertible section J ∈ C ∞ ( U r , Λ( C n − ) (cid:98) ⊗ End( D )) of degree 0 suchthat(5.18) ∂J ∂z + (cid:16) i ∂∂z B (cid:17) J = 0 . Let φ ∈ C ∞ ,c ( C n , [0 , φ = 1 on U r , the support ofwhich is included in U r . It is enough to show that for | z (cid:48) | small enough,there is an invertible section j z (cid:48) ∈ C ∞ ( C , Λ( C n − ) (cid:98) ⊗ End( D )) of totaldegree 0, which depends smoothly on z (cid:48) , such that(5.19) ∂j z (cid:48) ∂z + φ (cid:16) i ∂∂z B (cid:17) j z (cid:48) = 0 . If j z (cid:48) = 1 + g z (cid:48) , equation (5.19) is equivalent to(5.20) (cid:18) ∂∂z + φi ∂∂z B (cid:19) g z (cid:48) + φi ∂∂z B = 0 . The form dz iπz on C is locally integrable, and by Poincar´e-Lelong(2.25), we have the equation of currents, ∂ dz iπz = δ . (5.21)Let π , π be the two canonical projections C → C . Let π : C → C be given by ( z , z (cid:48) ) = z + z (cid:48) . Let C c ( C , C ) be the vector spaceof continuous complex functions on C with compact support. If f ∈ C c ( C , C ), put(5.22) Rf = π ∗ (cid:20) π ∗ f dz π ∗ dz iπz (cid:21) . Equation (5.22) gives the convolution of two distributions. Let C b ( C , C )be the vector space of bounded continuous functions from C into itself.Then Rf ∈ C b ( C , C ). By (5.21), we get(5.23) ∂∂z Rf = f. Let C ,b (cid:0) C , Λ ( C n − ) (cid:98) ⊗ End ( D ) (cid:1) be the space of bounded continuoussections of degree 0 of Λ ( C n − ) (cid:98) ⊗ End ( D ). Let F z (cid:48) be the bounded BISMUT , SHU
SHEN , AND ZHAOTING
WEI operator acting on C ,b (cid:0) C , Λ( C n − ) (cid:98) ⊗ End( D ) (cid:1) ,(5.24) F z (cid:48) = Rφi ∂∂z B . Since B is of degree 1, F z (cid:48) is of degree 0.By (5.23), we get ∂∂z F z (cid:48) = φi ∂∂z B . (5.25)By (5.22), (5.24), there is C > (cid:107) F z (cid:48) (cid:107) ≤ C (cid:13)(cid:13)(cid:13) B ( ≥ (cid:13)(cid:13)(cid:13) C b ( U r ) . Using (5.25), equation (5.20) can be written in the form(5.27) ∂∂z [(1 + F z (cid:48) ) g z (cid:48) + F z (cid:48)
1] = 0 . We may and we will assume that (cid:13)(cid:13)(cid:13) B ( ≥ (cid:13)(cid:13)(cid:13) C b ( U r ) is small enough, sothat, when using (5.26), (cid:107) F z (cid:48) (cid:107) <
1. Then 1 + F z (cid:48) is invertible withuniformly bounded inverse.Put g z (cid:48) = − (1 + F z (cid:48) ) − F z (cid:48) . (5.28)Then g z (cid:48) is a solution of degree 0 of (5.27), and so it solves (5.20). Alsoit is obviously smooth in all variables. Moreover,(5.29) (cid:107) g z (cid:48) (cid:107) ≤ (cid:107) F z (cid:48) (cid:107) − (cid:107) F z (cid:48) (cid:107) . If (cid:13)(cid:13)(cid:13) B ( ≥ (cid:13)(cid:13)(cid:13) C b ( U r ) is small enough so that (cid:107) F z (cid:48) (cid:107) ≤ /
3, then (cid:107) g z (cid:48) (cid:107) ≤ / j z (cid:48) = 1 + g z (cid:48) is invertible.Assume that i ≥ J k , B k ) for k ≤ i −
1. By (5.16), we have to find J i such that J i and ∂J i + B i − J i do not contain dz , . . . , dz i . This is equivalent to the fact that J i isholomorphic in z , . . . , z i − , and ∂J i ∂z i + (cid:16) i ∂∂zi B i − (cid:17) J i = 0 . (5.30)By the argument given in (5.16)–(5.29), using again a dilation δ t , | t | < (cid:13)(cid:13)(cid:13) B ( ≥ i − (cid:13)(cid:13)(cid:13) C b ( U r ) is small enough, so that (5.30)can be solved. Since (cid:0) ∂ + B i − (cid:1) = 0 and since B i − does not con-tain dz , . . . , dz i − , B i − is a holomorphic function of z , . . . , z i − . Thearguments that were used before show that the solution J i is also holo-morphic in z , . . . , z i − . OHERENT SHEAVES AND RRG 33
As explained after (5.16), for r > U = U r , thereexists a smooth section v of End ( D ) such that(5.31) J − A E , (cid:48)(cid:48) | U J = ∂ + v . Let J (0) be the component of J of degree 0 in Λ (cid:0) T ∗ X (cid:1) . By (5.6),(5.31), we get(5.32) (cid:2) J (0) (cid:3) − v J (0) = v . Put K = J (cid:2) J (0) (cid:3) − , ∇ D | U (cid:48)(cid:48) = J (0) ∂ (cid:2) J (0) (cid:3) − . (5.33)Then ∇ D | U (cid:48)(cid:48) is a holomorphic structure on D | U . By (5.31)–(5.33), weget(5.34) K − A E (cid:48)(cid:48) K = ∇ D | U (cid:48)(cid:48) + v , which completes the proof of our theorem. (cid:3) Antiholomorphic superconnections and coherent sheafs.
Let F be a smooth vector bundle on X , and let O ∞ X ( F ) be the sheafof smooth sections of F . Definition 5.4.
Let E be the sheaf of O X -complexes (cid:0) O ∞ X ( E ) , A E (cid:48)(cid:48) (cid:1) ,and let H E denote its cohomology. For r ≥
0, let ( E r , d r ) denotethe spectral sequence of O X -sheaves associated with the filtration by O ∞ X (cid:0) Λ (cid:0) T ∗ X (cid:1)(cid:1) .Let HD be the cohomology of ( O ∞ X ( D ) , v ).The sheaf H E is a Z -graded filtered sheaf of O X -modules. Also( E , d ) = (cid:0) O ∞ X (cid:0) Λ (cid:0) T ∗ X (cid:1) (cid:98) ⊗ D (cid:1) , v (cid:1) , E = O ∞ X (cid:0) Λ (cid:0) T ∗ X (cid:1)(cid:1) (cid:98) ⊗ O ∞ X HD. (5.35)The differential d on E is such that d = 0. Also it verifies Leibnizrule with respect to multiplication by Ω , • ( X, C ). Therefore d inducesa holomorphic structure on the O ∞ X -module HD , which will also bedenoted ∇ HD (cid:48)(cid:48) . Remark . Assume that HD has locally constant rank. Then HD isa smooth vector bundle on X . Then ∇ HD (cid:48)(cid:48) defines a canonical holo-morphic structure on HD .When fixing a splitting in (5.2), using (5.3), and writing A E (cid:48)(cid:48) as in(5.6), from the three equations in (5.7), we can also derive a construc-tion of ∇ HD (cid:48)(cid:48) . Indeed, by (5.7), the connection ∇ D (cid:48)(cid:48) induces a holo-morphic structure on the O ∞ X -module HD , which is just d = ∇ HD (cid:48)(cid:48) . BISMUT , SHU
SHEN , AND ZHAOTING
WEI
This holomorphic structure is canonical, and does not depend on thesplitting in (5.2), as is also clear from (5.11).
Definition 5.6.
Let H HD be the cohomology of the complex of O X -modules (cid:0) O ∞ X (cid:0) Λ (cid:0) T ∗ X (cid:1)(cid:1) (cid:98) ⊗ O ∞ X HD, ∇ HD (cid:48)(cid:48) (cid:1) .Then H HD is a Z -graded O X -module. By (5.35),(5.36) E = H HD.
We recall a fundamental result of Block [Bl10, Lemma 4.5].
Theorem 5.7.
The complex E has Z -graded coherent cohomology H E .The spectral sequence E r degenerates at E = H E , and F H E = 0 .For i ≥ , (5.37) H i HD = 0 . We have the identity of Z -graded coherent sheaves, (5.38) H E = H HD, and also the identity of O ∞ X -modules, (5.39) H E ⊗ O X O ∞ X = HD.
Proof.
By Theorem 5.3, if U is sufficiently small, after conjugation bya smooth section of Aut ( E ) on U , we may and we will assume thaton U , A E (cid:48)(cid:48) has the form(5.40) A E (cid:48)(cid:48) = v + ∇ D (cid:48)(cid:48) . In (5.40), ∇ D (cid:48)(cid:48) is a holomorphic structure on D | U that preserves the Z -grading and is such that(5.41) ∇ D (cid:48)(cid:48) v = 0 . In (5.40), A E (cid:48)(cid:48) now preserves the decreasing filtration associated withthe degree in D , which was not the case in the original form of A E (cid:48)(cid:48) .We will compute H E using the associated spectral sequence.Let H D be the cohomology of ( O U ( D ) , v ). By [D09, TheoremsII.3.13 and II.3.14], H D is a coherent sheaf. Using the Poincar´e lemmafor Dolbeault cohomology [D09, Lemma I.3.29], if V is a small polydisk,then(5.42) H E ( V ) = H D ( V ) , so that(5.43) H E = H D. Therefore
H E is a Z -graded coherent sheaf. Also, the natural filtration F is trivial on H E , so that F H E = 0, a property which is intrinsicto
H E , and does not depend on the choice of a conjugation.
OHERENT SHEAVES AND RRG 35
By [Ma67, Corollary VI.1.12], O ∞ X is flat on O X , so that the map(5.44) ρ : H D ⊗ O U O ∞ U → HD is an isomorphism of O ∞ U -modules. Moreover, the action of d on HD is just ∇ D (cid:48)(cid:48) . Using the Poincar´e lemma, we deduce that for i ≥ H i HD = 0, and also that(5.45) H D = H HD.
By the above, the cohomology of ( E , d ) coincides with H D = H E .Therefore, the original spectral sequence degenerates at E , and E = H E . Using (5.43), (5.44), we get (5.39). The proof of our theorem iscompleted. (cid:3)
Remark . Assume that HD has locally constant rank. As we saw inRemark 5.5, HD is equipped with a canonical holomorphic structure ∇ HD (cid:48)(cid:48) . By Theorem 5.7, we get(5.46) H E = O X ( HD ) . In general, HD depends only on D, v , and H HD depends only on D, v , ∇ D (cid:48)(cid:48) . By (5.38), H E depends only on
D, v , ∇ D (cid:48)(cid:48) , and not onthe full A E (cid:48)(cid:48) .5.4. The determinant line bundle.
Since
H E = H HD is a Z -graded coherent sheaf, by Knudsen-Mumford [KnM76], its determinantdet H E is a canonically defined holomorphic line bundle on X .Let det D denote the smooth determinant line bundle,(5.47) det D = r (cid:48) (cid:79) r (cid:0) det D i (cid:1) ( − i . Now we fix identifications in (5.2), (5.3), and we write A E (cid:48)(cid:48) as in(5.6). The antiholomorphic connection ∇ D (cid:48)(cid:48) on D induces a corre-sponding antiholomorphic connection ∇ det D (cid:48)(cid:48) on det D . Theorem 5.9.
The antiholomorphic connection ∇ det D (cid:48)(cid:48) on det D doesnot depend on the splitting in (5.2), (5.3). It defines a holomorphicstructure on det D . Finally, we have the canonical isomorphism ofholomorphic line bundles on X , (5.48) det D (cid:39) det H E . Proof.
We use the notation in (5.10), (5.11). Let ∇ det D (cid:48)(cid:48) denote theantiholomorphic connection on det D associated with ∇ D . By (5.11),we obtain(5.49) ∇ det D (cid:48)(cid:48) = ∇ det D (cid:48)(cid:48) + Tr s [[ α, v ]] . BISMUT , SHU
SHEN , AND ZHAOTING
WEI
Since supertraces vanish on supercommutators, by (5.49), we get(5.50) ∇ det D (cid:48)(cid:48) = ∇ det D (cid:48)(cid:48) , which proves the first part of our proposition. Also(5.51) ∇ det D (cid:48)(cid:48) , = Tr s (cid:2) ∇ D (cid:48)(cid:48) , (cid:3) . Using the third identity in (5.7), (5.51), and the fact that supertracesvanish on supercommutators, we get(5.52) ∇ det D (cid:48)(cid:48) , = 0 . so that ∇ det D (cid:48)(cid:48) is a holomorphic structure on det D .Now we use the notation in the proof of Theorem 5.7. On a smallopen set U in X , after a local conjugation by a smooth section ofAut ( E ), we can write A E (cid:48)(cid:48) in the form (5.40), ∇ D (cid:48)(cid:48) being now aholomorphic structure on D | U . By Knudsen-Mumford [KnM76], on U , the holomorphic line bundle det H E is just the holomorphic linebundle det D equipped with the holomorphic structure induced by thisspecific ∇ D (cid:48)(cid:48) . As we saw before, this is just our original ∇ det D (cid:48)(cid:48) whoseconstruction does not depend on the conjugation. The proof of ourtheorem is completed. (cid:3) The case where
H E = 0 . We will now give conditions underwhich
H E = 0.
Theorem 5.10.
The O X -module H E vanishes if and only if thereexists a smooth section k of degree − of End ( D ) such that (5.53) [ v , k ] = 1 . An equivalent condition is that for any x ∈ X , the complex ( D, v ) x isexact.Another equivalent condition is that there exists a smooth section k of End ( E ) of degree − such that (5.54) (cid:2) A E (cid:48)(cid:48) , k (cid:3) = 1 . Proof.
By Theorem 5.7,
H E = 0 if and only if we have the identity of O ∞ X -modules HD = 0. Since D is soft, by [D09, Proposition IV.4.14],this is equivalent to the exactness of the complex ( C ∞ ( X, D ) , v ).Since C ∞ ( X, D ) is projective over C ∞ ( X, C ), this is equivalent tothe existence of a smooth section k of End ( D ) of degree − v , k ] = 1, which implies that for any x ∈ X , the complex ( D, v ) x is exact. Conversely, if this last assumption is verified, we find easilythere is a global smooth k such that [ v , k ] = 1, from which we deducethat HD = 0. OHERENT SHEAVES AND RRG 37
To complete the proof of our theorem, we need to show that if
H E =0, then k exists. We use a splitting as in (5.2), (5.3), we replace E by E , and we write A E (cid:48)(cid:48) as in (5.6). We write k in the form(5.55) k = (cid:88) i ≥ k i , with k i a smooth section of Λ i (cid:0) T ∗ X (cid:1) (cid:98) ⊗ End ( D ) of degree −
1. Wechoose k as before.If α is a smooth section of Λ (cid:0) T ∗ X (cid:1) (cid:98) ⊗ End ( D ), we denote by α ( ≤ i ) the sum of the components of α whose degree in Λ (cid:0) T ∗ X (cid:1) is ≤ i , andby α ( i ) the component that lies in Λ i (cid:0) T ∗ X (cid:1) (cid:98) ⊗ End ( D ).Assume that we found k j , j ≤ i such that(5.56) (cid:34) A E (cid:48)(cid:48) , (cid:88) j ≤ i k j (cid:35) ( ≤ i ) = 1 . We will show how to construct k i +1 so that (5.56) is solved with i replaced by i + 1. Assuming that (5.56) has been solved, the corre-sponding equation with i replaced by i + 1 can be written in the form,(5.57) [ v , k i +1 ] + (cid:34) A E (cid:48)(cid:48) , (cid:88) j ≤ i k j (cid:35) ( i +1) = 0 . Because of (5.56), we get(5.58) (cid:34) v , (cid:34) A E (cid:48)(cid:48) , (cid:88) j ≤ i k j (cid:35)(cid:35) ( i +1) = (cid:34) A E (cid:48)(cid:48) , (cid:34) A E (cid:48)(cid:48) , (cid:88) j ≤ i k j (cid:35)(cid:35) ( i +1) = 0 . By (5.58), to solve (5.57), we can take(5.59) k i +1 = − k , (cid:34) A E (cid:48)(cid:48) , (cid:88) j ≤ i k j (cid:35) ( i +1) . This completes the proof of the existence of k .Another more direct proof is as follows. The vector bundle End ( E )can be equipped with the antiholomorphic superconnection ad (cid:0) A E (cid:48)(cid:48) (cid:1) .Moreover, 1 is a section of End ( E ) such that(5.60) (cid:2) A E (cid:48)(cid:48) , (cid:3) = 0 . We claim that the complex (cid:0) C ∞ ( X, End ( E )) , ad (cid:0) A E (cid:48)(cid:48) (cid:1)(cid:1) is exact, whichimplies the existence of k . Indeed let L ( k ) denote left multiplication BISMUT , SHU
SHEN , AND ZHAOTING
WEI by k acting on End ( D ). Then(5.61) [ad ( v ) , L ( k )] = 1 , so that the complex (End ( D ) , ad ( v )) is exact. Consider the globalspectral sequence associated with the filtration F . By (5.61), the firstterm of the spectral sequence vanishes identically, so that the complex (cid:0) C ∞ ( X, End ( E )) , ad (cid:0) A E (cid:48)(cid:48) (cid:1)(cid:1) is exact. The proof of our theorem iscompleted. (cid:3) Superconnections, morphisms, and cones.
Let (cid:0)
E, A E (cid:48)(cid:48) (cid:1) beanother couple similar to (cid:0) E, A E (cid:48)(cid:48) (cid:1) . The objects associated with E will be underlined. A morphism φ : E → E is a smooth morphismof degree 0 of Z -graded Λ (cid:0) T ∗ X (cid:1) -vector bundles, which is such that A E (cid:48)(cid:48) φ = φA E (cid:48)(cid:48) . Then φ induces a morphism of O X -complexes E → E that commutes with multiplication by O ∞ X (cid:0) Λ (cid:0) T ∗ X (cid:1)(cid:1) . Then φ inducesa morphism E → E , a morphism of O ∞ X -complexes ( D, v ) → ( D, v ),a morphism of O X -complexes (cid:0) HD, ∇ HD (cid:48)(cid:48) (cid:1) → (cid:0) HD, ∇ HD (cid:48)(cid:48) (cid:1) , and mor-phisms of coherent sheaves H E → H E .Here, we follow the conventions of Subsection 4.3. We form the cone(5.62) C = cone ( E , E ) . By (4.26), the underlying vector bundle C is such that(5.63) C • = E • +1 ⊕ E • . Then C is an object similar to E . The corresponding antiholomorphicsuperconnection is denoted A C (cid:48)(cid:48) φ , and as in (4.27), we have the identity,(5.64) A C (cid:48)(cid:48) φ = (cid:20) A E (cid:48)(cid:48) φ ( − deg A E (cid:48)(cid:48) . (cid:21) As in (4.28), we have the exact sequence of complexes,(5.65) 0 (cid:47) (cid:47) E • (cid:47) (cid:47) C • (cid:47) (cid:47) E • +1 (cid:47) (cid:47) , from which we get the long exact sequence of O X -modules,(5.66) . . . (cid:47) (cid:47) H • E (cid:47) (cid:47) H • C (cid:47) (cid:47) H • +1 E φ ( − · +1 (cid:47) (cid:47) H • +1 E . . . . We have corresponding exact sequences involving the O ∞ X -modules HD and their pointwise version.If φ : H E → H E is an isomorphism, then φ is said to be a quasi-isomorphism. As we saw in Subsection 4.3, this is equivalent to(5.67) H C = 0 . Using Theorem 5.10, we find that φ is a quasi-isomorphism if and onlyif for any x ∈ X , φ : D x → D x is a quasi-isomorphism. OHERENT SHEAVES AND RRG 39
Given t ∈ C , we can still define M t ∈ End ( C ) as in (4.30). Namely, M t acts by multiplication by 1 on E , and by multiplication by t on E .As in (4.31), for t (cid:54) = 0, we have the identity(5.68) A C (cid:48)(cid:48) tφ = M t A C (cid:48)(cid:48) φ M − t . Pull-backs.
Let Y be a compact complex manifold, and let f : X → Y be a holomorphic map. If H is a vector bundle on Y , f ∗ H denote the pull-back of H to X . Then df ∗ maps f ∗ T ∗ Y into T ∗ X .Therefore Λ (cid:0) T ∗ X (cid:1) is a Λ (cid:0) f ∗ T ∗ Y (cid:1) -module.We use the notation of Subsection 5.1, except that now (cid:0) F, A F (cid:48)(cid:48) (cid:1) isan antiholomorphic superconnection on Y with diagonal bundle D F .Put(5.69) E = Λ (cid:0) T ∗ X (cid:1) (cid:98) ⊗ Λ ( f ∗ T ∗ Y ) f ∗ F. Then E is a Z -graded vector bundle on X , the degree in E being thesum of the degrees in the two factors of the right hand-side. Also E is a Λ (cid:0) T ∗ X (cid:1) -module, so that it is equipped with the correspondingfiltration.By (4.6), we get(5.70) E p,q = Λ p (cid:0) T ∗ X (cid:1) (cid:98) ⊗ f ∗ D qF . In particular, E on X has exactly the same properties as F on Y , andthe associated diagonal bundle is D E = f ∗ D F .Let µ be the canonical morphism f ∗ F → E . Then µ induces acorresponding map C ∞ ( Y, F ) → C ∞ ( X, E ) which is such that if α ∈ Ω , • ( Y, C ) , s ∈ C ∞ ( Y, F ), then(5.71) µ ( αs ) = df ∗ ( α ) µs. Proposition 5.11.
There is a unique antiholomorphic superconnection A E (cid:48)(cid:48) on E such that if s ∈ C ∞ ( Y, F ) , then (5.72) A E (cid:48)(cid:48) ( µs ) = µ (cid:0) A F (cid:48)(cid:48) s (cid:1) . Proof.
Let α ∈ Ω , • ( Y, C ). By (5.4), (5.71), we get(5.73) µ (cid:0) A F (cid:48)(cid:48) αs (cid:1) = ∂ X ( df ∗ α ) µ ( s ) + ( − deg α ( df ∗ α ) µ (cid:0) A F (cid:48)(cid:48) s (cid:1) . In particular, if df ∗ α = 0, (5.73) vanishes. This completes the proof ofour proposition. (cid:3) We will use the notation(5.74) E = f ∗ b F. The notation emphasizes the fact that f ∗ b F is not the classical pull-back f ∗ F . BISMUT , SHU
SHEN , AND ZHAOTING
WEI
Similarly, if E = (cid:0) E, A E (cid:48)(cid:48) (cid:1) , F = (cid:0) F, A F (cid:48)(cid:48) (cid:1) , we use the notation(5.75) E = f ∗ b F . There is an associated morphism of Z -graded O X -modules O X ⊗ f − O Y f − H F → H E .If φ : F → F (cid:48) is a morphism, it induces a morphism f ∗ b φ : f ∗ b F → f ∗ b F (cid:48) .We fix a smooth splitting as in (5.2), that induces the non-canonicalidentification F (cid:39) F in (5.3). This choice induces a correspondingidentification E (cid:39) E . Also, µ induces a canonical map f ∗ F → E .We write A F (cid:48)(cid:48) as in (5.9).The connection ∇ D F (cid:48)(cid:48) induces a connection ∇ f ∗ D F (cid:48)(cid:48) on f ∗ D F . Let µB be the section of Λ (cid:0) T ∗ X (cid:1) (cid:98) ⊗ End ( f ∗ D F ) deduced from B by theabove pull-back map. Then(5.76) A E (cid:48)(cid:48) = ∇ f ∗ D F (cid:48)(cid:48) + µB. If α ∈ Ω , • ( Y, C ), from now on, we will also use the notation f ∗ α instead of df ∗ α .5.8. Tensor products.
Let E = (cid:0) E, A E (cid:48)(cid:48) (cid:1) , F = (cid:0) F, A F (cid:48)(cid:48) (cid:1) be two an-tiholomorphic superconnections on X , with associated diagonal vectorbundles D E , D F . Put(5.77) E (cid:98) ⊗ b F = E (cid:98) ⊗ Λ ( T ∗ X ) F. This defines an antiholomorphic superconnection E (cid:98) ⊗ b F = (cid:16) E (cid:98) ⊗ b F, A E (cid:98) ⊗ b F (cid:48)(cid:48) (cid:17) .The corresponding diagonal vector bundle is D E (cid:98) ⊗ D F .6. An equivalence of categories
The purpose of this Section is to establish a result of Block [Bl10],who showed there is an equivalence of categories between the homotopycategory of antiholomorphic superconnections on X and D bcoh ( X ).This Section is organized as follows. In Subsection 6.1, if F is anobject in D bcoh ( X ), we construct a corresponding O X -complex of O ∞ X -modules F ∞ , and we show it is quasi-isomorphic to F .In Subsection 6.2, we introduce the dg-category B dg ( X ) of the anti-holomorphic superconnections E , the 0-cycle category B ( X ), and thehomotopy category B ( X ). In the next Subsections, we construct anequivalence of categories between B ( X ) and D bcoh ( X ).In Subsection 6.3, we show that the natural functor F X : B ( X ) → D bcoh ( X ) is essentially surjective.In Subsection 6.4, we describe in more detail the homotopy categoryB ( X ). OHERENT SHEAVES AND RRG 41
In Subsection 6.5, we show that the induced functor F X = B ( X ) → D bcoh ( X ) is an equivalence of categories.In Subsection 6.6, we show the above equivalence of categories iscompatible with pull-backs.In Subsection 6.7, we prove the compatibility of this equivalence withtensor products.Finally, in Subsection 6.8, we consider the case of direct images.6.1. Some properties of D bcoh ( X ) . Let F ∈ D bcoh ( X ), and let d F beits differential. Set(6.1) F ∞ = O ∞ X ⊗ O X F . Then F ∞ is a perfect O ∞ X -complex. Let d F ∞ be the differential on F ∞ , and let H F ∞ be the cohomology of F ∞ . Since O ∞ X is flat over O X [Ma67, Corollary VI.1.12], we have(6.2) H F ∞ = O ∞ X ⊗ O X H F . Put(6.3) F ∞ = O ∞ X (cid:0) Λ (cid:0) T ∗ X (cid:1)(cid:1) (cid:98) ⊗ O X F . Since F is a complex of O X -modules, we can equip F ∞ with thedifferential d F ∞ given by(6.4) d F ∞ = ∂ X + d F . By Poincar´e Lemma and by a theorem of Malgrange [Ma67, Corol-lary VI.1.12], we have a quasi-isomorphism of O X -complexes,(6.5) F → F ∞ . Equivalently
H F , H F ∞ denote the corresponding cohomology sheaves,we have the canonical isomorphism of O X -modules,(6.6) H F = H F ∞ . Also F ∞ is equipped with the filtration induced by Λ (cid:0) T ∗ X (cid:1) . Let F ∞ r , r ≥ O X -modules.Then(6.7) F ∞ = O ∞ X (cid:0) Λ (cid:0) T ∗ X (cid:1)(cid:1) (cid:98) ⊗ O ∞ X H F ∞ . Also F ∞ inherits a differential d , which can be viewed as a holomor-phic structure ∇ H F ∞ (cid:48)(cid:48) on H F ∞ . Let H F ∞ denote the cohomologyof (cid:16) F ∞ , d (cid:17) . Then(6.8) F ∞ = H F ∞ . BISMUT , SHU
SHEN , AND ZHAOTING
WEI
Proposition 6.1.
The spectral sequence F ∞ r degenerates at F ∞ , and F H F ∞ = 0 .For i ≥ , (6.9) H i F ∞ = 0 . We have the identity of Z -graded coherent sheaves, (6.10) H F = H F ∞ = H F ∞ . Proof.
Using the special form of d F ∞ in (6.4), the proof is essentiallythe same as the proof of Theorem 5.7. It is left to the reader. (cid:3) Three categories.
Let X be a compact complex manifold. If E = (cid:0) E, A E (cid:48)(cid:48) (cid:1) is an antiholomorphic superconnection on X , put(6.11) E X = C ∞ ( X, E ) . Let E = (cid:0) E, A E (cid:48)(cid:48) (cid:1) , E = (cid:0) E, A E (cid:48)(cid:48) (cid:1) be antiholomorphic supercon-nections on X . Let Hom ( E, E ) be the vector bundle of Λ (cid:0) T ∗ X (cid:1) -morphisms from E into E . Then Hom ( E, E ) is a Z -graded vectorbundle. If k ∈ Z , Hom k ( E, E ) consists of the morphisms that in-crease the degree by k . Then Hom ( E, E ) is also equipped with theinduced antiholomorphic superconnection A Hom(
E,E ) (cid:48)(cid:48) . We use the no-tation Hom ( E , E ) = (cid:0) Hom (
E, E ) , A Hom(
E,E ) (cid:48)(cid:48) (cid:1) .Following [Ke06, Subsection 2.2], we define B dg ( X ) to be the dg-category whose objects are the E X , and the morphisms Hom ( E , E ) X .Let B( X ) = Z (B ( X )) be the 0-cycle category associated withB dg ( X ). Its objects coincide with the objects of B dg ( X ), and if E , E are taken as before, its morphisms are given by(6.12) Z Hom ( E , E ) X = (cid:8) φ ∈ Hom ( E , E ) X , A Hom(
E,E ) (cid:48)(cid:48) φ = 0 (cid:9) . Also E , E are said to be isomorphic if there exists φ ∈ Z Hom ( E X , E X ) , ψ ∈ Z Hom ( E X , E X ) such that ψφ = 1 E X , φψ = 1 E X . (6.13)Let B( X ) be the homotopy category associated with B( X ). Its ob-jects coincide with the objects of B( X ), and the morphisms are givenby the morphisms of H Hom ( E , E ) X , the cohomology group of degree0 associated with the complex (cid:0) Hom ( E , E ) X , A Hom(
E,E ) (cid:48)(cid:48) (cid:1) .The aim of this section is to establish an equivalence of categoriesbetween B( X ) and D bcoh ( X ). The proof consists of the following threesteps:(1) Construct an essentially surjective functor F X : B( X ) → D bcoh ( X ). OHERENT SHEAVES AND RRG 43 (2) Prove that the above functor factors though a functor F X :B( X ) → D bcoh ( X ).(3) Show that F X is fully faithful.6.3. Essential surjectivity.
Let E = (cid:0) E, A E (cid:48)(cid:48) (cid:1) be an antiholomor-phic superconnection. By Theorem 5.7, E ∈ D bcoh ( X ). We obtain thisway a functor F X : B ( X ) → D bcoh ( X ). Theorem 6.2.
The functor F X is essentially surjective, i.e., if F ∈ D bcoh ( X ) , there is E ∈ B( X ) and an isomorphism F X ( E ) (cid:39) F in D bcoh ( X ) .Proof. The proof of our theorem is divided into the next four steps. (cid:3)
The following result is established by Illusie in [Col71, PropositionII.2.3.2]. We give a proof for completeness.
Proposition 6.3.
There exists a bounded complex of finite dimensionalcomplex smooth vector bundles (cid:0)
Q, d Q (cid:1) and a quasi-isomorphism of O ∞ X -complexes φ : O ∞ X Q → F ∞ .Proof. We may and we will assume that for i < F i = 0. Put(6.14) k = sup (cid:8) i ≥ , H i F (cid:54) = 0 (cid:9) . We will show our result by induction on k , while also proving that wemay take Q i = 0 for i > k . • The case k = 0Assume first that k = 0. Since H F is a coherent sheaf, if x ∈ X , there exist an open neighborhood U of x , a bounded holomorphiccomplex R U of trivial holomorphic vector bundles in nonpositive degreeon U , and a holomorphic morphism r U : O U R U → H F | U such thatwe have the exact sequence of O U -modules,(6.15) 0 (cid:47) (cid:47) O U R − (cid:96) U U (cid:47) (cid:47) . . . (cid:47) (cid:47) O U R U r U (cid:47) (cid:47) H F | U (cid:47) (cid:47) . Since O ∞ U is flat over O U , we get the exact sequence of O ∞ U -modules(6.16)0 (cid:47) (cid:47) O ∞ U R − (cid:96) U U (cid:47) (cid:47) . . . (cid:47) (cid:47) O ∞ U R U r U (cid:47) (cid:47) H F ∞ | U (cid:47) (cid:47) . Since X is compact, there is a finite cover U of X by such U . Set (cid:96) = sup U ∈U (cid:96) U . (6.17)We will establish our result by induction on (cid:96) . If (cid:96) = 0, then H F is locally free, i.e., there is a smooth holomorphic bundle F on X such that H F = O X F . We consider Q = F as a trivial complex BISMUT , SHU
SHEN , AND ZHAOTING
WEI concentrated in degree 0 and r : O ∞ X Q → F ∞ to be the canonicalinjection.Assume now that (cid:96) ≥
1, and that our proposition holds if the corre-sponding length defined in (6.17) is ≤ (cid:96) −
1. From the exact sequence(6.16), we get a morphism of complexes O ∞ U R U → F ∞ | U . Put(6.18) C U = cone ( O ∞ U R U , F ∞ | U ) . Since k = 0, C U is exact. Observe that C U is just the complex(6.19)0 (cid:47) (cid:47) O ∞ U R − (cid:96) U U (cid:47) (cid:47) . . . (cid:47) (cid:47) O ∞ U R U r U (cid:47) (cid:47) F , ∞ | U (cid:47) (cid:47) F , ∞ (cid:47) (cid:47) . Since R U is trivial on U , we can extend R U to a trivial vector bundleon X . Set R = (cid:77) U ∈U R U . (6.20)Then, R is a trivial vector bundle on X , which we identify with a trivialcomplex in degree 0. Also we have an obvious morphism of complexes R | U → R U .Let ( ϕ U | U ∈U ) be a smooth partition of unity subordinated to U . Put(6.21) r = (cid:88) U ∈U ϕ U r U . Then r defines a morphism of O ∞ X -complexes O ∞ X R → F ∞ . Moreover,for any U , r induces surjections r | U : O ∞ U R → H F ∞ | U .We claim that r | U lifts to a smooth morphism of complexes R | U → R U such that the following diagram commutes: O ∞ U R Ur U (cid:15) (cid:15) O ∞ U R r | U (cid:47) (cid:47) (cid:58) (cid:58) F ∞ | U . (6.22)Let us prove this claim. Since F ∞ | U is a subcomplex of C U , fromthe morphism r | U : O ∞ U R → F ∞ | U , we get a corresponding morphism s U : O ∞ U R → C U , which is also clear by (6.19).Since C U is exact and since O ∞ X R is free and concentrated in degree 0,by [BGS88c, Lemma 3.3], there is a null-homotopy h U : O ∞ U R • → C •− U such that(6.23) s U = d C U h U . Inspection of (6.19) shows that h U provides the desired lift in (6.22). OHERENT SHEAVES AND RRG 45
Since r U is a quasi-isomorphism, by (6.22) and [KaS90, Proposition1.4.4], we get a quasi-isomorphism of O ∞ U -complexes,(6.24) O ∞ U cone( R | U , R U ) → cone( O ∞ U R, F ∞ | U ) . We use the notation R U , R | U to designate the sheaves O ∞ U R U , O ∞ U R .By proceeding as in (6.19), we can rewrite (6.24) as a quasi-isomorphismof complexes of O ∞ U -modules,0 (cid:47) (cid:47) R − (cid:96) U U (cid:47) (cid:47) (cid:15) (cid:15) · · · (cid:47) (cid:47) (cid:15) (cid:15) R − U ⊕ R | U γ (cid:47) (cid:47) (cid:15) (cid:15) R U (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) (cid:47) (cid:47) · · · (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) R | U (cid:47) (cid:47) F , ∞ | U (cid:47) (cid:47) F , ∞ | U (cid:47) (cid:47) · · · , (6.25)so that the corresponding bicomplex is exact. By construction, thecohomology of the second row in (6.25) is concentrated in degree − H − . This forces γ to be a surjective morphism of O ∞ U -modules. Since R U and R are free, for any x ∈ U , γ x is surjective,and ker γ is a sheaf defined by a smooth vector bundle K U on U . Since U is contractible, K U is a trivial vector bundle, so that the complex(6.26) 0 (cid:47) (cid:47) R − (cid:96) U U (cid:47) (cid:47) · · · (cid:47) (cid:47) R − U (cid:47) (cid:47) ker γ → H − . Since here (cid:96) is replaced here by (cid:96) −
1, by ourinduction argument, there exists a bounded complex of finite dimen-sional complex vector bundles (cid:0) Q (cid:48) , d Q (cid:48) (cid:1) with Q (cid:48) i = 0 if i ≥
0, such thatif Q (cid:48) = O ∞ X Q (cid:48) , we have a quasi-isomorphism of O ∞ X -complexes,(6.27) Q (cid:48) → cone( R , F ∞ ) . By (4.28), we have a canonical morphism cone • ( R , F ∞ ) → R • +1 . Weobtain a tautological diagram(6.28) Q (cid:48)• (cid:47) (cid:47) (cid:38) (cid:38) cone • ( R , F ∞ ) (cid:15) (cid:15) R • +1 . Since the horizontal morphism in (6.28) is a quasi-isomorphism, by[KaS90, Proposition 1.4.4 (TR5)], we get a quasi-isomorphism of O ∞ X -complexes(6.29) cone( Q (cid:48)• , R • +1 ) → cone(cone • ( R , F ∞ ) , R • +1 ) . Put Q • = cone (cid:0) Q (cid:48)•− , R • (cid:1) , Q • = cone( Q (cid:48) •− , R • ) , (6.30) BISMUT , SHU
SHEN , AND ZHAOTING
WEI so that Q = O ∞ X Q . Observe that for i > Q i = 0.By [KaS90, Proposition 1.4.4 (TR3)], we have a homotopy equiva-lence of O ∞ X -complexes(6.31) cone(cone •− ( R , F ∞ ) , R • ) → F ∞ . By (6.29)–(6.31), we get a quasi-isomorphism,(6.32) Q → F ∞ , which completes the proof of our theorem when k = 0. • The case k ≥ k ≥ k (cid:48) ≤ k −
1. Considerthe truncated complex(6.33) τ ≤ k − F ∞ : 0 (cid:47) (cid:47) F , ∞ (cid:47) (cid:47) · · · (cid:47) (cid:47) F k − , ∞ (cid:47) (cid:47) ker d F ∞ | F k − , ∞ (cid:47) (cid:47) . Then τ ≤ k − F ∞ is a subcomplex of F ∞ . The cohomology of τ ≤ k − F ∞ is given by H i F ∞ , ≤ i ≤ k −
1. By our induction argument, thereexists a complex Q of complex vector bundles on X such that Q i = 0for i ≥ k , and that if Q = O ∞ X Q , there exists a quasi-isomorphism(6.34) Q → τ ≤ k − F ∞ . By composition with the inclusion τ ≤ k − F ∞ → F ∞ , we get a mor-phism of complex(6.35) Q → F ∞ . The cohomology of cone( Q , F ∞ ) is concentrated in degree k .Let d be the differential of cone( Q , F ∞ ). Consider the truncatedcomplex(6.36) τ ≥ k cone( Q , F ∞ ) : 0 → cone k ( Q , F ∞ ) /d cone k − ( Q , F ∞ ) → cone k +1 → . . . By our induction hypothesis, there exists a smooth complex of vectorbundles Q such that Q i = 0 for i > k , and that if Q = O ∞ X Q , wehave a quasi-isomorphism Q → τ ≥ k cone( Q , F ∞ ) . (6.37)By proceeding as in (6.22), (6.23) the morphism (6.37) lifts to a quasi-isomorphism Q → cone( Q , F ∞ ) . (6.38)Put Q = cone (cid:0) Q •− , Q • (cid:1) , Q = cone (cid:0) Q •− , Q • (cid:1) , (6.39) OHERENT SHEAVES AND RRG 47 so that Q = O ∞ X Q . Then Q i = 0 for i > k . By proceeding as in(6.27)-(6.32), from (6.38), we get a quasi-isomorphism Q → F ∞ , (6.40)which completes the proof of our proposition. (cid:3) We take
Q, φ as in Proposition 6.3. Put(6.41) C = cone ( Q , F ∞ ) . As in (4.26)–(4.28), we have the identity(6.42) C • = Q • +1 ⊕ F ∞• , the differential d C is given by(6.43) d C = (cid:20) d Q φ ( − deg d F ∞ (cid:21) , and we have the exact sequence of complexes,(6.44) 0 → F ∞• → C • → Q • +1 → . Since φ is a quasi-isomorphism, C is exact.As in (6.3), if A is a Z -graded O ∞ X -module, we use the notation(6.45) A = O ∞ X (cid:0) Λ (cid:0) T ∗ X (cid:1)(cid:1) (cid:98) ⊗ A . Put(6.46) Q = Λ (cid:0) T ∗ X (cid:1) (cid:98) ⊗ Q. Then Q is the O ∞ X -module of smooth sections of the vector bundle Q .Since O ∞ X (cid:0) Λ (cid:0) T ∗ X (cid:1)(cid:1) is flat over O ∞ X , φ : Q → F ∞ (6.47)is still a quasi-isomorphism, and we have the exact sequence(6.48) 0 → F ∞• → C • → Q • +1 → . Also C is exact.Put(6.49) R = Hom (cid:16) Q • +1 , C • (cid:17) . Let d R be the differential on R which comes from d Q , d C . Also thesupercommutator with d C is a differential on End (cid:0) C (cid:1) .Let A ∈ End (cid:0) C (cid:1) be a lower triangular. We write A as a matrixwith respect to the splitting (6.42),(6.50) A = (cid:20) α β γ (cid:21) . BISMUT , SHU
SHEN , AND ZHAOTING
WEI
Put(6.51) A = (cid:20) αβ (cid:21) . Then A ∈ R . Proposition 6.4.
The following identity holds: (6.52) (cid:104) d C , A (cid:105) = d R A − ( − deg( A ) (cid:20) γφ ( − deg (cid:21) . Proof.
We only need to establish (6.52) when α = 0 , β = 0, which iseasy. (cid:3) As before, we denote by R X the global sections of R on X . Proposition 6.5.
The complexes R and R X are exact.Proof. We know that C is exact. Also Q is locally free. It is now easyto see that R is exact. Since R is soft, the functor R → R X is exact,and we obtain the corresponding result for R X . (cid:3) Remark . For a related argument, we refer to [St20, Tag 0647].We will establish a fundamental result of Block [Bl10, Lemma 4.6].
Theorem 6.7.
Given F ∈ D bcoh ( X ) , there exists E = (cid:0) E, A E (cid:48)(cid:48) (cid:1) ∈ B( X ) and a morphism of O ∞ X (cid:0) Λ (cid:0) T ∗ X (cid:1)(cid:1) -modules φ : E → F ∞ , whichis a quasi-isomorphism of O X -complexes, and induces a quasi-isomorphismof O ∞ X -complexes ( D, v ) → F ∞ . In particular, E and F are isomor-phic in D bcoh ( X ) .Proof. We use the notation introduced before. We will reformulate ourproblem as being the construction of a generalized antiholomorphicsuperconnection d C on C which will have the structure(6.53) d C = (cid:20) A E (cid:48)(cid:48) φ ( − deg d F ∞ (cid:21) . such that if(6.54) d C = (cid:88) k ∈ N d C k is the canonical expansion of d C with respect to Λ (cid:0) T ∗ X (cid:1) , then(6.55) d C = d C . By (6.4), for k ≥ d F ∞ k = 0 . OHERENT SHEAVES AND RRG 49
By (6.53), we get(6.57) d C k = (cid:20) v k φ k ( − deg d F ∞ k (cid:21) . In particular, for k = 0, we get v = d Q , d F ∞ = d F ∞ . (6.58)For i ≥
0, put(6.59) d C ≤ i = i (cid:88) k =0 d C k . We will use a similar notation for other related expressions. We willconstruct d C i by induction so that for i ≥ (cid:104) d C ≤ i (cid:105) ≤ i = 0 . For i = 0, we take d C = d C . For i ≥
0, we will show how to construct d C ≤ i +1 from d C ≤ i . We have the identity(6.61) d C ≤ i +1 = d C ≤ i + d C i +1 . Then(6.62) (cid:104) d C ≤ i +1 (cid:105) ≤ i +1 = (cid:104) d C ≤ i , d C i +1 (cid:105) ≤ i +1 + (cid:104) d C ≤ i (cid:105) ≤ i +1 . Clearly,(6.63) (cid:104) d C ≤ i , d C i +1 (cid:105) ≤ i +1 = (cid:104) d C , d C i +1 (cid:105) . For (6.62) to vanish, using (6.63), we should have(6.64) (cid:104) d C , d C i +1 (cid:105) + (cid:104) d C ≤ i (cid:105) ≤ i +1 = 0 . By (6.60), (6.64) can also be written in the form(6.65) (cid:104) d C , d C i +1 (cid:105) + (cid:104) d C ≤ i (cid:105) i +1 = 0 . Since d F ∞ , = 0 , we have the identity(6.66) (cid:104) d F ∞ ≤ i +1 (cid:105) ≤ i +1 = 0 . By (6.56), the expansion of d F ∞ terminates at k = 1. Here, this fact will beignored. BISMUT , SHU
SHEN , AND ZHAOTING
WEI
By (6.65), (6.66), we deduce that(6.67) (cid:104) d F ∞ , d F ∞ i +1 (cid:105) + (cid:104) d F ∞ ≤ i (cid:105) i +1 = 0 . First, we solve (6.65) for i = 0. Namely, we need to find d C suchthat(6.68) (cid:104) d C , d C (cid:105) = 0 . By Proposition 6.4, equation (6.68) can be written in the form(6.69) d R d C + (cid:20) d F ∞ φ ( − deg (cid:21) = 0 . Recall that by (6.58), v = d Q . Then(6.70) d R (cid:20) d F ∞ φ ( − deg (cid:21) = (cid:20) d F ∞ d F ∞ φ ( − deg − d F ∞ φ ( − deg v (cid:21) . Since (cid:104) d F ∞ , d F ∞ (cid:105) = 0, and d F ∞ φ = φ v , and since v is odd, by(6.70), we get(6.71) d R (cid:20) d F ∞ φ ( − deg (cid:21) = 0 . Let ∇ Q (cid:48)(cid:48) be an antiholomorphic connection on Q . Put e R = d C − (cid:20) ∇ Q (cid:48)(cid:48) (cid:21) , f R = d R (cid:20) ∇ Q (cid:48)(cid:48) (cid:21) + (cid:20) d F ∞ φ ( − deg (cid:21) . (6.72)An elementary computation shows that(6.73) f R = (cid:20) ∇ Q (cid:48)(cid:48) v φ ( − deg ∇ Q (cid:48)(cid:48) + d F ∞ φ ( − deg (cid:21) . Differentiation on X having disappeared, e R , f R lie in R X .By (6.71), (6.72), solving equation (6.69) is equivalent to finding e R ∈ R X such that(6.74) d R e R + f R = 0 . By (6.71), (6.72), we get(6.75) d R f R = 0 . By Proposition 6.5, the complex R X is exact. Using (6.75), there is asolution e R to (6.74), which gives the existence of a solution for (6.68). OHERENT SHEAVES AND RRG 51
Now we assume that i ≥
1. We will still find a solution for (6.65).By Proposition 6.4, we get(6.76) (cid:104) d C , d C i +1 (cid:105) = d R d C i +1 + (cid:20) d F ∞ i +1 φ ( − deg (cid:21) . By (6.65), (6.76), we obtain,(6.77) d R d C i +1 + (cid:104) d C ≤ i (cid:105) i +1 + (cid:20) d F ∞ i +1 φ ( − deg (cid:21) = 0 . Using (6.60), we get(6.78) (cid:20) d C , (cid:104) d C ≤ i (cid:105) i +1 (cid:21) = (cid:20) d C , (cid:104) d C ≤ i (cid:105) (cid:21) i +1 = (cid:104) d C ≤ i , d C , ≤ i (cid:105) i +1 = 0 . By Proposition 6.4 and by (6.78), we get(6.79) d R (cid:104) d C ≤ i (cid:105) i +1 − (cid:34) (cid:104) d F ∞ ≤ i (cid:105) i +1 φ ( − deg (cid:35) = 0 . An easy computation shows that(6.80) d R (cid:20) d F ∞ i +1 φ ( − deg (cid:21) = (cid:20) d F ∞ d F ∞ i +1 φ ( − deg − d F ∞ i +1 φ ( − deg v (cid:21) . Since φ is a morphism of complexes, and since v is odd, using (6.67),we can rewrite (6.80) in the form(6.81) d R (cid:20) d F ∞ i +1 φ ( − deg (cid:21) = − (cid:34) (cid:104) d F ∞ , ≤ i (cid:105) i +1 φ ( − deg (cid:35) By combining (6.79) and (6.81), we get(6.82) d R (cid:20)(cid:104) d C ≤ i (cid:105) i +1 + (cid:20) d F ∞ i +1 φ ( − deg (cid:21)(cid:21) = 0 . Since the complex R X is exact, using (6.82), equation (6.77) for d C i +1 can be solved.In summary, we have constructed E = ( E, A E (cid:48)(cid:48) ) ∈ B( X ) and amorphism of O X -complexes φ : E → F ∞ . Therefore φ induces amorphism(6.83) φ : H E → H F ∞ . By construction, φ induces an isomorphism of O ∞ X -modules,(6.84) HD (cid:39) H F ∞ . BISMUT , SHU
SHEN , AND ZHAOTING
WEI
Both sheaves in (6.84) are equipped with antiholomorphic connections,which necessarily correspond by φ . By combining equation (5.38) inTheorem 5.7 and equation (6.10) in Proposition 6.1, we deduce that in(6.83), φ is an isomorphism. The proof of our theorem is completed. (cid:3) The homotopy category.
Let E , E be two objects in B( X ).Note that Hom ( E , E ) is an object in B ( X ). Recall that Z Hom ( E , E ) X was defined in (6.12). A morphism φ ∈ Z Hom ( E , E ) X is said to benull-homotopic if there exists ψ ∈ Hom − ( E , E ) X such that(6.85) φ = A Hom(
E,E ) (cid:48)(cid:48) ψ. Also φ ∈ Z Hom ( E , E ) X is said to be a homotopy equivalence ifthere is ψ ∈ Z Hom ( E , E ) X such that ψφ − | E is null-homotopicin Z Hom ( E , E ) X , and φψ − | E is null-homotopic in Z Hom ( E , E ) X .If φ ∈ Z Hom ( E , E ) X is null-homotopic, the image of this morphismin C bcoh ( X ) is also null-homotopic. In particular, the functor F X :B( X ) → D bcoh ( X ) factorizes through a functor F X : B( X ) → D bcoh ( X ).By Theorem 6.2, F X is essentially surjective. Proposition 6.8. If φ ∈ Z Hom ( E , E ) X , the following conditions areequivalent: (1) φ is a homotopy equivalence, (2) φ induces a quasi-isomorphism E → E , (3) For any x ∈ X , φ induces a quasi-isomorphism D x → D x . (4) φ induces a quasi-isomorphism O ∞ X ( D ) → O ∞ X ( D ) . (5) φ induces a homotopy equivalence C ∞ ( X, D ) → C ∞ ( X, D ) .Proof. This is an easy consequence of Theorem 5.10 and of the con-struction of cone ( E , E ). (cid:3) An equivalence of categories.
We have a fundamental theoremof Block [Bl10, Theorem 4.3].
Theorem 6.9.
The functor F X : B( X ) → D bcoh ( X ) is an equivalenceof triangulated categories.Proof. Since F X is essentially surjective, we only need to show that F X is fully faithful. Equivalently, if E , E ∈ B ( X ), we have to show that thecorresponding map Hom B( X ) ( E , E ) → Hom D bcoh (X) ( E , E ) is bijective.Let us prove surjectivity. A morphism in Hom D bcoh (X) ( E , E ) is repre-sented by the diagram of morphisms of O X -complexes, E F (cid:47) (cid:47) q . i . (cid:111) (cid:111) E , (6.86) OHERENT SHEAVES AND RRG 53 where F ∈ D bcoh ( X ), and ‘q.i.’ stands for ‘quasi-isomorphism’. Alsothere are obvious maps O ∞ X Λ (cid:0) T ∗ X (cid:1) (cid:98) ⊗ O X E → E , O ∞ X Λ (cid:0) T ∗ X (cid:1) (cid:98) ⊗ O X E → E . By (6.86), we obtain the commutative diagram, E F q . i . (cid:111) (cid:111) (cid:15) (cid:15) (cid:47) (cid:47) EF ∞ (cid:96) (cid:96) (cid:62) (cid:62) (6.87)By (6.87), we get the diagram of O X -complexes, E F ∞ (cid:47) (cid:47) q . i . (cid:111) (cid:111) E , (6.88)with O ∞ X Λ (cid:0) T ∗ X (cid:1) -morphisms. By Theorem 6.7, there exists E ∈ B( X )and a quasi-isomorphism E → F ∞ . We can replace (6.88) by E E (cid:47) (cid:47) q . i . (cid:111) (cid:111) E , (6.89)the morphisms in (6.89) being morphisms of B( X ). By Proposition6.8, the first morphism in (6.89) is a homotopy equivalence. Using itshomotopic inverse, we get the desired morphism from E → E .We now prove injectivity. Let φ , φ : E → E be two morphisms inB( X ), whose images coincide in D bcoh ( X ). By definition, there is F ∈ D bcoh ( X ) and a quasi-isomorphism φ : F → E , such that φ φ = φ φ upto homotopy in Hom C bcoh (X) ( F , E ).Using again Theorem 6.7, there exists E ∈ B ( X ) and a quasi-isomorphism E → F ∞ . Therefore, in the above construction, we mayas well replace F by E , so that φ is a quasi-isomorphism E → E , and φ φ = φ φ up to homotopy in Z Hom (cid:0) E , E (cid:1) X . Since φ is a quasi-isomorphism, this is a homotopy equivalence, so that φ = φ in B ( X ),which completes the proof of our theorem. (cid:3) Compatibility with pull-backs.
We make the same assump-tions as in Subsection 5.7 and we use the corresponding notation. Bythe results of that Subsection, we have defined a pull-back functor f ∗ b : B ( Y ) → B ( X ). Recall that the definition of the left-derivedfunctor Lf ∗ : D bcoh ( Y ) → D bcoh ( X ) was given in Subsection 3.2. BISMUT , SHU
SHEN , AND ZHAOTING
WEI
Proposition 6.10.
The following diagram commutes up to isomor-phism: (6.90) B ( Y ) F Y (cid:47) (cid:47) f ∗ b (cid:15) (cid:15) D bcoh ( Y ) Lf ∗ (cid:15) (cid:15) B ( X ) F X (cid:47) (cid:47) D bcoh ( X ) Proof. If E is an object in B ( Y ), if E = F Y ( E ), then E is an object inD bcoh ( Y ). Also f ∗ E is given by (3.4). Since E is a bounded complex offlat O Y -modules, by (3.5), we have Lf ∗ E = f ∗ E . (6.91)By Proposition 5.11, there is a morphism of O X -complexes µ : f ∗ E → f ∗ b E . Using (3.5), to establish our proposition, we only need to provethat if x ∈ X , the morphism µ induces a quasi-isomorphism of thecorresponding stalks at x .Take x ∈ X, y = f ( x ) ∈ Y . By Theorem 5.3, to establish ourtheorem, we may and we will assume that on a small open neighborhood V of y , D is a trivial holomorphic vector bundle, E = Λ (cid:0) T ∗ V (cid:1) (cid:98) ⊗ D ,and A E (cid:48)(cid:48) | V = v + ∇ D (cid:48)(cid:48) . Put U = f − V . By proceeding as in the proofof Theorem 5.7, we get H f ∗ E = H ( f ∗ O Y ( D ) , f ∗ v ) , H f ∗ b E | U = H ( O U f ∗ D, f ∗ v | U ) . (6.92)Also note that(6.93) f − O Y ⊗ f − O Y O X = O X . Since D is trivial on V , by (6.93), we get(6.94) f − O Y ( D ) ⊗ f − O Y O X | U = O X ( f ∗ D ) | U . By (6.92), (6.94), we obtain(6.95) H f ∗ E = H f ∗ b E . This completes the proof that µ is a quasi-isomorphism. (cid:3) Compatibility with tensor products.
Let E , F be objects inB ( X ). We use the notation of Subsections 3.3 and 5.7. Proposition 6.11.
We have a canonical isomorphism, (6.96) F X (cid:0) E (cid:98) ⊗ b F (cid:1) (cid:39) F X ( E ) (cid:98) ⊗ L O X F X ( F ) in D bcoh ( X ) . OHERENT SHEAVES AND RRG 55
Proof.
We will identify E , F with their images by F X in D bcoh ( X ).Since E , F are flat over O X , by (3.10), we get(6.97) E (cid:98) ⊗ L O X F = E (cid:98) ⊗ O X F . There is a natural morphism φ : E (cid:98) ⊗ O X F → E (cid:98) ⊗ b F of objects inD bcoh ( X ). We will show that it is a quasi-isomorphism.We write E , F in the form, E = (cid:0) E, A E (cid:48)(cid:48) (cid:1) , F = (cid:0) F, A F (cid:48)(cid:48) (cid:1) . (6.98)Let D E , D F be the diagonal vector bundles associated with E , F , andlet v ,E , v ,F be the corresponding endomorphisms of D E , D F . The ob-jects associated with E (cid:98) ⊗ b F are denoted in the same way. In particular, D E (cid:98) ⊗ b F = D E (cid:98) ⊗ D F , v ,E (cid:98) ⊗ b F = v ,E (cid:98) ⊗ (cid:98) ⊗ v ,F . (6.99)By Theorem 5.3, if U ⊂ X is a small open set, after conjugation, wemay assume that A E (cid:48)(cid:48) | U = v ,E + ∇ D E | U (cid:48)(cid:48) , A F (cid:48)(cid:48) | U = v ,F + ∇ D F | U (cid:48)(cid:48) . (6.100)In particular D E | U , D F | U are holomorphic vector bundles, so that D E (cid:98) ⊗ b F | U is also a holomorphic vector bundle. Let ∇ D E (cid:98) ⊗ bF | U (cid:48)(cid:48) be the correspond-ing holomorphic structure. By (6.100), we get(6.101) A E (cid:98) ⊗ b F (cid:48)(cid:48) | U = v ,E (cid:98) ⊗ b F + ∇ D E (cid:98) ⊗ bF | U (cid:48)(cid:48) . By Theorem 5.7, the morphisms ( O U ( D E ) , v ,E ) → E | U , ( O U ( D F ) , v ,F ) → F | U are quasi-isomorphisms. Since E | U and O U ( D E ) are flat over O U ,and the same is true for the objects associated with F , the obviousmorphism(6.102) ( O U ( D E ) , v ,E ) (cid:98) ⊗ O U ( O U ( D F ) , v ,F ) → E (cid:98) ⊗ O U F is a quasi-isomorphism.Observe that(6.103)( O U ( D E ) , v ,E ) (cid:98) ⊗ O U ( O U ( D F ) , v ,F ) = (cid:0) O U (cid:0) D E (cid:98) ⊗ b F (cid:1) , v ,E (cid:98) ⊗ b F (cid:1) . It is now clear that φ is a quasi-isomorphism of objects in D bcoh ( X ).The proof of our proposition is completed. (cid:3) Direct images.
Let Y be a compact complex manifold, and let f : X → Y be a holomorphic map.Let E be an object in B ( X ), so that F X ( E ) is an object in D bcoh ( X ).Since E is a bounded complex of soft O ∞ X -modules, by (3.11), we get(6.104) Rf ∗ E = f ∗ E . BISMUT , SHU
SHEN , AND ZHAOTING
WEI
Also f ∗ maps O ∞ Y (cid:0) Λ (cid:0) T ∗ Y (cid:1)(cid:1) to f ∗ O ∞ X (cid:0) Λ (cid:0) T ∗ X (cid:1)(cid:1) . The differential of f ∗ E verifies Leibniz’s rule with respect to multiplication by O ∞ Y (cid:0) Λ (cid:0) T ∗ Y (cid:1)(cid:1) ,so that f ∗ E is a complex of O ∞ Y Λ (cid:0) T ∗ Y (cid:1) -modules. Proposition 6.12.
There exists E = (cid:0) E, A E (cid:48)(cid:48) (cid:1) ∈ B ( Y ) and a mor-phism of O ∞ Y (cid:0) Λ (cid:0) T ∗ Y (cid:1)(cid:1) -modules φ : E → f ∗ E , which is also a quasi-isomorphism of O Y -complexes.Proof. We use the notation in Subsection 6.1. In particular, as in (6.3),we get(6.105) f ∗ E ∞ = O ∞ Y (cid:0) Λ (cid:0) T ∗ Y (cid:1)(cid:1) (cid:98) ⊗ O Y f ∗ E . Also O ∞ Y (cid:0) Λ (cid:0) T ∗ Y (cid:1)(cid:1) acts on f ∗ E . We get this way a morphism of com-plexes ψ : f ∗ E ∞ → f ∗ E , which is also a morphism of O ∞ Y (cid:0) Λ (cid:0) T ∗ Y (cid:1)(cid:1) -modules. By (6.5), the map f ∗ E → f ∗ E ∞ is a quasi-isomorphism of O Y -complexes. These maps fit into the triangle(6.106) f ∗ E (cid:47) (cid:47) id (cid:34) (cid:34) f ∗ E ∞ ψ (cid:15) (cid:15) f ∗ E . From (6.106), we deduce that ψ is a quasi-isomorphism.By a theorem of Grauert [GrR84, Theorem 10.4.6], f ∗ E defines an ob-ject in D bcoh ( Y ). By Theorem 6.7, there exists an object E = (cid:0) E, A E (cid:48)(cid:48) (cid:1) in B ( Y ) and a morphism of O ∞ Y (cid:0) Λ (cid:0) T ∗ Y (cid:1)(cid:1) -modules ρ : E → f ∗ E ∞ ,which is a quasi-isomorphism of O Y -complexes. Then φ = ψρ : E → f ∗ E is the quasi-isomorphism we are looking for, which completes ourproof. (cid:3) Antiholomorphic superconnections and generalizedmetrics
In this Section, if (cid:0)
E, A E (cid:48)(cid:48) (cid:1) is an antiholomorphic superconnection,if E is equipped with a splitting so that E (cid:39) E , if h is a generalizedmetric on D , we define the adjoint superconnection A E (cid:48) of A E (cid:48)(cid:48) withrespect to h . This is just the obvious analogue of the construction of theholomorphic part of the Chern connection on a holomorphic Hermitianvector bundle, which we briefly reviewed in Subsection 2.4.This Section is organized as follows. In Subsection 7.1, we construct A E (cid:48) as the formal adjoint of A E (cid:48)(cid:48) with respect to a non-degenerateHermitian form θ h .In Subsection 7.2, we conclude the Section by introducing the cur-vature of the superconnection A E = A E (cid:48)(cid:48) + A E (cid:48) . OHERENT SHEAVES AND RRG 57
The adjoint of an antiholomorphic superconnection.
Wemake the same assumptions as in Subsection 5.1 and we use the corre-sponding notation.We define the antiautomorphism (cid:101) of Λ ( T ∗ C X ) as in Subsection 4.4,so that equation (4.33) holds. Also we define deg − on Λ ( T ∗ C X ) as in(4.38). If Ω ( X, C ) denote the vector space of smooth forms on X , wedenote by Ω − ,i ( X, C ) the vector space of the α ∈ Ω ( X, C ) such thatdeg − α = i .Here, we follow [B13, Section 6.6]. If α, α (cid:48) ∈ Ω ( X, C ), put(7.1) θ ( α, α (cid:48) ) = i n (2 π ) n (cid:90) X (cid:101) α ∧ α (cid:48) . One verifies easily that θ defines a non-degenerate Hermitian form onΩ ( X, C ), and that the Ω − ,i ( X, C ) are mutually orthogonal. If B ∈ Ω ( X, C ) acts on Ω ( X, C ) by exterior multiplication, we denote by B † the adjoint of B with respect to θ . An easy computation, which alsofollows from [B13, Proposition 6.5.1] shows that if f ∈ T ∗ C X , then(7.2) f † ∧ = − f ∧ By (7.2), we deduce that if B ∈ Ω ( X, C ), B † is just the ‘adjoint’ B ∗ we defined in Subsection 4.4.Since Λ ( T ∗ X ) is a Z -graded holomorphic vector bundle on X , usingthe results of Subsection 5.8, Λ ( T ∗ X ) (cid:98) ⊗ E has the same properties as E , i.e., it can be equipped with an antiholomorphic superconnection A Λ( T ∗ X ) (cid:98) ⊗ E (cid:48)(cid:48) , and it is also a Λ ( T ∗ C X )-module. If α ∈ Ω ( X, C ) , s ∈ C ∞ (cid:0) X, Λ ( T ∗ X ) (cid:98) ⊗ E (cid:1) , the obvious extension of (5.4) holds, i.e.,(7.3) A Λ( T ∗ X ) (cid:98) ⊗ E (cid:48)(cid:48) ( αs ) = (cid:16) ∂ X α (cid:17) s + ( − deg α αA Λ( T ∗ X ) (cid:98) ⊗ E (cid:48)(cid:48) s. In the sequel, we will still use the notation A E (cid:48)(cid:48) instead of A Λ( T ∗ X ) (cid:98) ⊗ E (cid:48)(cid:48) .We fix a splitting of E as in (5.2), (5.3), so that(7.4) Λ ( T ∗ X ) (cid:98) ⊗ E (cid:39) Λ ( T ∗ C X ) (cid:98) ⊗ D. Put(7.5) Ω (
X, D ) = C ∞ (cid:0) X, Λ ( T ∗ C X ) (cid:98) ⊗ D (cid:1) . By (7.4), (7.5), we deduce that(7.6) Ω (
X, D ) = C ∞ (cid:0) X, Λ ( T ∗ X ) (cid:98) ⊗ E (cid:1) . We equip Ω (
X, D ) with the total degree associated with the degreedeg − on Ω ( X, C ) and with the degree in D . Let Ω − ,i ( X, D ) be thevector space of the s ∈ Ω (
X, D ) such that deg s = i .In the sequel, we write A E (cid:48)(cid:48) instead of A Λ( T ∗ X ) (cid:98) ⊗ E (cid:48)(cid:48) . BISMUT , SHU
SHEN , AND ZHAOTING
WEI
Definition 7.1.
Put E ∗ i = (cid:77) p + q = i Λ p (cid:0) T ∗ X (cid:1) ⊗ D q ∗ , E ∗ = r (cid:48) (cid:77) i = r E i ∗ . (7.7)Then(7.8) C ∞ (cid:0) X, Λ ( T ∗ X ) (cid:98) ⊗ E ∗ (cid:1) = Ω ( X, D ∗ ) . If s ∈ Ω (
X, D ) , s (cid:48) ∈ Ω (cid:16) X, D ∗ (cid:17) , we can write s, s (cid:48) in the form s = (cid:88) α i r i , s (cid:48) = (cid:88) β j t j , (7.9)with α i , β j ∈ Ω ( X, C ), and r i ∈ C ∞ ( X, D ) , t j ∈ C ∞ (cid:16) X, D ∗ (cid:17) . Definition 7.2.
Put(7.10) θ ( s, s (cid:48) ) = i n (2 π ) n (cid:88) (cid:90) X (cid:101) α i ∧ β j (cid:10) r i , t j (cid:11) . Note that given i, j , the expression in the right-hand side of (7.10)is nonzero if and only if deg − α i = deg − β j , deg r i = deg t j , so thatdeg − α i deg r i = deg − β j deg t j .If s, s (cid:48) are given instead by s = (cid:88) r i α i , s (cid:48) = (cid:88) t j β j , (7.11)from (7.10), and from the previous considerations, we deduce that θ ( s, s (cid:48) ) is still given by the right-hand side of (7.10).If k is a smooth section of End ( D ), let (cid:101) k be its transpose, which isa section of End ( D ∗ ). It follows from the previous considerations thatthe adjoint of k with respect to θ is given by (cid:101) k , i.e., it is just the actionof the classical expected adjoint.We define the smooth bundle M D of generalized metrics on D as inSubsection 4.4. As in Subsection 4.4, we will say that a smooth section h of M D is pure if h = h .Let h be a smooth section of M D . Then h defines an invertiblemorphism from Ω ( X, D ) into Ω (cid:16)
X, D ∗ (cid:17) . Definition 7.3. If s, s (cid:48) ∈ Ω (
X, D ), put(7.12) θ h ( s, s (cid:48) ) = θ ( s, hs (cid:48) ) . Then θ h is a non-degenerate Hermitian form on Ω ( X, D ), and theΩ − ,i ( X, D ) are mutually orthogonal with respect to θ h . OHERENT SHEAVES AND RRG 59
Definition 7.4.
Let A E (cid:48) denote the formal adjoint of A E (cid:48)(cid:48) with re-spect to θ h .Then A E (cid:48) verifies Leibniz rule when replacing ∂ X by ∂ X , and it isof degree −
1. Also(7.13) A E (cid:48) , = 0 . Let A E ∗ (cid:48) denote the formal adjoint of A E (cid:48)(cid:48) with respect to θ . Then(7.14) A E (cid:48) = h − A E ∗ (cid:48) h. The above constructions are compatible with the conventions in[BL95, Subsections 1 (c) and 1 (d)] and in [B13, Section 3.5]. In [Q16,Proposition 2.17], Qiang followed the same lines to define the adjoint A E (cid:48) when h is a pure metric.Let ∇ D (cid:48) be the adjoint of ∇ D (cid:48)(cid:48) with respect to θ h . If i = 0 , or i ≥ v ∗ i be the adjoint of v i with respect to θ h . By (5.6), we obtain(7.15) A E (cid:48) = v ∗ + ∇ D (cid:48) + (cid:88) i ≥ v ∗ i . Recall that B was defined in (5.8). Its adjoint B ∗ is given by(7.16) B ∗ = v ∗ + (cid:88) i ≥ v ∗ i . Then equation (7.15) can be written in the form(7.17) A E (cid:48) = ∇ D (cid:48) + B ∗ . Let ∇ D (cid:48) be the adjoint of ∇ D (cid:48)(cid:48) when h is replaced by the puremetric h . Then there is a smooth section γ of total degree − ( ≥ ( T ∗ C X ) (cid:98) ⊗ End ( D ) such that(7.18) ∇ D (cid:48) = ∇ D (cid:48) + γ. Put(7.19) ∇ D = ∇ D (cid:48)(cid:48) + ∇ D (cid:48) . Then ∇ D is a classical unitary connection on D with respect to h .The Hermitian metric h on D induces a classical Hermitian metric h det D on det D . Let ∇ det D be the connection on det D that is inducedby ∇ D . Then ∇ det D is a unitary connection with respect to h det D . ByTheorem 5.9, ∇ det D (cid:48)(cid:48) is a holomorphic structure on det D . In general, ∇ D (cid:48)(cid:48) does not define a holomorphic structure on D , so that ∇ D isnot a Chern connection. BISMUT , SHU
SHEN , AND ZHAOTING
WEI
Proposition 7.5.
The connection ∇ det D coincides with the Chern con-nection on (cid:0) det D, h det D (cid:1) .Proof. Let ∇ det D (cid:48) be the holomorphic connection on det D that is in-duced by ∇ D (cid:48) . By construction, ∇ det D (cid:48) is the adjoint of ∇ det D (cid:48)(cid:48) withrespect to h det D , which gives our proposition. (cid:3) Curvature.
We make the same assumptions as in Subsection 7.1,and we use again the superconnection formalism of Quillen [Qu85].Let A E be the superconnection on E ,(7.20) A E = A E (cid:48)(cid:48) + A E (cid:48) . Set ∇ D = ∇ D (cid:48)(cid:48) + ∇ D (cid:48) , C = B + B ∗ . (7.21)By (5.9), (7.17), (7.20), and (7.21), we get(7.22) A E = ∇ D + C. The curvature of A E is given by(7.23) A E , = (cid:2) A E (cid:48)(cid:48) , A E (cid:48) (cid:3) . Then A E , is a smooth section of Λ ( T ∗ C X ) (cid:98) ⊗ End ( D ) of degree 0. Also,it verifies Bianchi’s identities, (cid:2) A E (cid:48)(cid:48) , A E , (cid:3) = 0 , (cid:2) A E (cid:48) , A E , (cid:3) = 0 . (7.24) Proposition 7.6.
The following identity holds: (7.25) (cid:2) A E , (cid:3) ∗ = A E , . Proof.
This is the obvious consequence of the fact that A E , (cid:48) = (cid:2) A E (cid:48)(cid:48) (cid:3) ∗ . (cid:3) Generalized metrics and Chern character forms
In this Section, given an antiholomorphic superconnection and a gen-eralized metric, we construct the corresponding Chern character forms,and we prove that their Bott-Chern class does not depend on the met-ric. Also, we show that the Chern character extends to D bcoh ( X ).As we explained in the introduction, some results which are provedin the present Section were already stated by Qiang in [Q16, Q17]. Wewill refer to them in more detail in the text.This Section is organized as follows. In Subsection 8.1, we constructthe Chern character forms, and we establish their main properties.In Subsection 8.2, we give a trivial example involving K¨ahler forms,that will be of special importance in Sections 13–16. OHERENT SHEAVES AND RRG 61
In Subsection 8.3, we describe the behavior of the Chern characterforms under pull-backs.In Subsection 8.4, we evaluate the Chern character of a tensor prod-uct.In Subsection 8.5, we consider the case where
H E is locally free.In Subsection 8.6, we study the behaviour of the Chern characterform under a suitable scaling of the metric. Also we show that indegree (1 , BC : D bcoh ( X ) → H (=)BC ( X, R ).In Subsection 8.9, we show that ch BC factors through K ( X ), i.e., itinduces a map K ( X ) → H (=)BC ( X, R ).Finally, in Subsection 8.10, we introduce spectral truncations forantiholomorphic superconnections. The results obtained there will beused in an infinite-dimensional context in Section 11.We make the same assumptions as in Sections 5 and 7, and we usethe corresponding notation.8.1. The Chern character forms.
We fix a square root of i = √− ϕ denote the mor-phism of Λ ( T ∗ C X ) that maps α to (2 iπ ) − deg α/ α .Recall that by Proposition 4.3, there is a well defined supertrace Tr s :Λ ( T ∗ X ) (cid:98) ⊗ End ( E ) → Λ ( T ∗ C X ), that vanishes on supercommutators.Here, we may as well view Tr s as a map from Λ ( T ∗ C X ) (cid:98) ⊗ End ( D ) toΛ ( T ∗ C X ).Let (cid:0) E, A E (cid:48)(cid:48) (cid:1) be an antiholomorphic superconnection on X . We fixsplittings as in (5.2), (5.3). Let h ∈ M D be a generalized metric, andlet A E be the associated superconnection defined in (7.20). Definition 8.1.
Set(8.1) ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) = ϕ Tr s (cid:2) exp (cid:0) − A E , (cid:1)(cid:3) . Then ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) is a smooth form on X .Let d M D denote the de Rham operator on M D . Then h − d M D h isa 1-form on M D with values in morphisms in (cid:2) Λ ( T ∗ X ) (cid:98) ⊗ End ( E ) (cid:3) that are self-adjoint with respect to h .Recall that the classical metric h on D was defined in (4.39).We will extend the result of Bott-Chern [BoC65, Proposition 3.28]described in Proposition 2.3, and results of Quillen [Qu85, Section BISMUT , SHU
SHEN , AND ZHAOTING
WEI
Theorem 8.2.
The form ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) lies in Ω (=) ( X, R ) , it is closed,and its Bott-Chern cohomology class does not depend on h . More pre-cisely, ϕ Tr s (cid:104) h − d M D h exp (cid:0) − A E , (cid:1)(cid:105) is a -form on M D with valuesin Ω (=) ( X, R ) , and moreover, (8.2) d M D ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) = − ∂ X ∂ X iπ ϕ Tr s (cid:104) h − d M D h exp (cid:0) − A E , (cid:1)(cid:105) . If g is a smooth section of Aut ( E ) , the Bott-Chern cohomologyclass of ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) is unchanged when replacing A E (cid:48)(cid:48) by gA E (cid:48)(cid:48) g − .In particular, the Bott-Chern cohomology class of ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) does notdepend on the splitting in (5.2), and depends only on A E (cid:48)(cid:48) .Finally, (8.3) ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) ≤ = ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) ≤ . Proof.
Let N D ∈ End ( D ) count the degree in D , and let N Λ ( T ∗ C X ) − count the degree in Λ ( T ∗ C X ) as the difference of the antiholomorphicand the holomorphic degrees. Since A E , has total degree 0, for t > e tN Λ ( T ∗ C X ) − exp (cid:0) − A E , (cid:1) e − tN Λ ( T ∗ C X ) − = e − tN D exp (cid:0) − A E , (cid:1) e tN D . Since e − tN D is an even operator, and since supertraces vanish on su-percommutators, by (8.1), (8.4), we get(8.5) e tN Λ ( T ∗ C X ) − ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) = ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) , which just says that ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) is a form in Ω (=) ( X, C ).Using the Bianchi identities (7.24) and the fact that supertraces van-ish on supercommutators, we get ∂ X Tr s (cid:2) exp (cid:0) − A E , (cid:1)(cid:3) = 0 , ∂ X Tr s (cid:2) exp (cid:0) − A E , (cid:1)(cid:3) = 0 , (8.6)i.e., the above form is closed. Since A E , is self-adjoint, it is elementaryto verify that ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) is a real form . In the proof, we use explicitly the fact that h is of degree 0. OHERENT SHEAVES AND RRG 63
We denote by [ ] + the anticommutator. Since exp (cid:0) − A E , (cid:1) is even,we get(8.7)Tr s (cid:104) h − d M D h exp (cid:0) − A E , (cid:1)(cid:105) = 12 Tr s (cid:20)(cid:104) h − d M D h, exp (cid:0) − A E , (cid:1)(cid:105) + (cid:21) . The 1-form (cid:104) h − d M D h, exp (cid:0) − A E , (cid:1)(cid:105) + takes its values in h self-adjointsections of Λ ( T ∗ C X ) (cid:98) ⊗ End ( D ) and is of degree 0. By the same con-siderations as before, we deduce that the form obtained from (8.7) bynormalization by ϕ lies in Ω (=) ( X, R ).In the sequel, we use (7.23) repeatedly. Also, we make 1-forms on M D to anticommute with odd forms on X . Clearly,(8.8) (cid:104) d M D , A E (cid:48) (cid:105) = − (cid:104) A E (cid:48) , h − d M D h (cid:105) , so that(8.9) d M D (cid:2) A E (cid:48)(cid:48) , A E (cid:48) (cid:3) = (cid:104) A E (cid:48)(cid:48) , (cid:104) A E (cid:48) , h − d M D h (cid:105)(cid:105) . By (8.9), we get(8.10) d M D Tr s (cid:2) exp (cid:0) − A E , (cid:1)(cid:3) = − Tr s (cid:104)(cid:104) A E (cid:48)(cid:48) , (cid:104) A E (cid:48) , h − d M D h (cid:105)(cid:105) exp (cid:0) − A E , (cid:1)(cid:105) . Using the Bianchi identities (7.24) and (8.10), we get (8.2), from whichwe deduce that the Bott-Chern cohomology class of ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) doesnot depend on h .Let g be a smooth section of Aut ( E ), and let h be a smooth sectionof M D . Let g ∗ be the adjoint of g with respect to h . The adjoint of gA E (cid:48)(cid:48) g − with respect to h is given by g ∗− A E (cid:48) g ∗ , so that the corre-sponding curvature is given by (cid:2) gA E (cid:48)(cid:48) g − , g ∗− A E (cid:48) g ∗ (cid:3) . Observe that(8.11) (cid:2) gA E (cid:48)(cid:48) g − , g ∗− A E (cid:48) g ∗ (cid:3) = g (cid:2) A E (cid:48)(cid:48) , ( g ∗ g ) − A E (cid:48) g ∗ g (cid:3) g − . Also ( g ∗ g ) − A E (cid:48) g ∗ g is just the adjoint of A E (cid:48)(cid:48) with respect to themetric g † h as defined in (4.40). By (8.11), since g is of degree 0, we get(8.12) ch (cid:0) gA E (cid:48)(cid:48) g − , h (cid:1) = ch (cid:0) A E (cid:48)(cid:48) , g † h (cid:1) . By (8.12), we deduce that the Bott-Chern cohomology class of ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) is unchanged when replacing A E (cid:48)(cid:48) by gA E (cid:48)(cid:48) g − .When changing the splitting in equation (5.2), as explained at theend of Subsection 4.1, we obtain a smooth section g of Aut ( E ), andthe new antiholomorphic superconnection on E associated with the Our definition of g ∗ is different from the one in (4.40). BISMUT , SHU
SHEN , AND ZHAOTING
WEI fixed A E (cid:48)(cid:48) is just gA E (cid:48)(cid:48) g − . Using the above results, it is now clearthat the Bott-Chern cohomology class of ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) does not dependon the splitting, and so it only depends on A E (cid:48)(cid:48) .Let v ∗ , , A E (cid:48) , A E denote the analogues of v ∗ , A E (cid:48)(cid:48) , A E when replac-ing h by the pure metric h . Proceeding as in the proof of (7.18), thereare smooth sections δ, (cid:15) of T ∗ C X (cid:98) ⊗ End ( D ) , Λ ( ≥ ( T ∗ C X ) (cid:98) ⊗ End ( D ) oftotal degree 0 , − A E (cid:48) = A E (cid:48) + (cid:2) δ, v ∗ , (cid:3) + (cid:15). By (8.13), we get(8.14) A E = A E + (cid:2) δ, v ∗ , (cid:3) + (cid:15). For (cid:96) ∈ R , let A E (cid:96) be obtained from A E by replacing (cid:15) by (cid:96)(cid:15) . AChern-Simons transgression shows that(8.15) ∂∂(cid:96) Tr s (cid:104) exp (cid:16) − A E , (cid:96) (cid:17)(cid:105) = − d X Tr s (cid:104) (cid:15) exp (cid:16) − A E, (cid:96) (cid:17)(cid:105) . Since (cid:15) is of degree ≥ T ∗ C X ), the right-hand side of (8.15) isof degree ≥
4, so that (8.15) vanishes in degree ≤
2. We see that indegree ≤ s (cid:2) exp (cid:0) − A E , (cid:1)(cid:3) = Tr s (cid:104) exp (cid:16) − A E , (cid:96) | (cid:96) =0 (cid:17)(cid:105) . For m ∈ R , let B E m be the analogue of A E in (8.14), in which δ, (cid:15) are replaced by mδ,
0. The same argument as before shows that(8.17) ∂∂m Tr s (cid:2) exp (cid:0) − B E , m (cid:1)(cid:3) = − d X Tr s (cid:2)(cid:2) δ, v ∗ , (cid:3) exp (cid:0) − B E, m (cid:1)(cid:3) . By (8.17), we deduce that(8.18) ∂∂m Tr s (cid:2) exp (cid:0) − B E, m (cid:1)(cid:3) ( ≤ = − d X Tr s (cid:2)(cid:2) δ, v ∗ , (cid:3) exp (cid:0) − (cid:2) v , v ∗ , (cid:3)(cid:1)(cid:3) . Since (cid:2) v ∗ , , (cid:2) v , v ∗ , (cid:3)(cid:3) = 0, we get(8.19) − Tr s (cid:2)(cid:2) δ, v ∗ , (cid:3) exp (cid:0) − (cid:2) v , v ∗ , (cid:3)(cid:1)(cid:3) = Tr s (cid:2)(cid:2) v ∗ , , δ exp (cid:0) − (cid:2) v , v ∗ , (cid:3)(cid:1)(cid:3)(cid:3) = 0 . By (8.19), (8.17) vanishes. Combining (8.16) and the vanishing of(8.17), we get (8.3). The proof of our theorem is completed. (cid:3)
Remark . The arguments of deformation over P given in [BGS88a,Section 1 f)] and in the proof of Proposition 2.3 can also be used toprove some of the results established in Theorem 8.2. In particular, toprove independence on the splitting of E , one can instead interpolatebetween two splittings of E over P and proceed as in [BGS88a]. OHERENT SHEAVES AND RRG 65
In [Q16, Proposition 2.24 and Corollary 3.14], Qiang established thatin the case where h is a pure metric, the form ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) is closed, andthat its Bott-Chern class does not depend on h . In [Q16, Subsection4.18], Qiang shows that the ch BC (cid:0) A E (cid:48)(cid:48) (cid:1) is invariant when conjugating A E (cid:48)(cid:48) . Definition 8.4.
Let ch BC (cid:0) A E (cid:48)(cid:48) (cid:1) ∈ H (=)BC ( X, R ) be the common Bott-Chern cohomology class of the forms ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) .8.2. A trivial example.
Here, we follow [B13, Remark 4.5.3]. Weconsider the trivial C X in Example 5.2, with D = C , and associatedantiholomorphic superconnection ∂ X . Let ω X be a smooth real (1 , iω X is a self-adjoint operator with respect tothe Hermitian form θ in (7.1). The exponential e − iω X in Λ ( T ∗ C X ) isalso self-adjoint. In particular h ω X = exp (cid:0) − iω X (cid:1) defines a generalizedmetric on C , i.e., it can be viewed as an element of M C . Using thenotation in (7.12), if s, s (cid:48) ∈ Ω ( X, C ), then(8.20) θ h ωX ( s, s (cid:48) ) = θ (cid:16) s, e − iω X s (cid:48) (cid:17) . The adjoint A C (cid:48) of A C (cid:48)(cid:48) is given by(8.21) A C (cid:48) = ∂ X − i∂ X ω X . If A C = A C (cid:48)(cid:48) + A C (cid:48) , the curvature A C , is given by(8.22) A C , = − ∂ X ∂ X iω X . By (8.22), we get(8.23) ch (cid:0) A C (cid:48)(cid:48) , h ω X (cid:1) = exp (cid:32) − ∂ X ∂ X iω X π (cid:33) . If we take ω X = 0, we get the trivial form 1.More generally, if we make the same assumptions as in Subsection8.1, if h ∈ M D , he − iω X lies also in M D . Even when h is pure, when ω X is nonzero, he − iω X is not pure.The above example will play a fundamental role in Section 13. Indeedthe theory of the hypoelliptic Laplacian in [B13] can be reinterpretedin terms of the above choice of a generalized metric on C . in which itis crucial that ω X is the K¨ahler form of a Hermitian metric on X .8.3. The Chern character of pull-backs.
Let Y be a compact com-plex manifold, and let f : X → Y be a holomorphic map as in Subsec-tion 5.7. Let (cid:0) F, A F (cid:48)(cid:48) (cid:1) be an antiholomorphic superconnection on Y , BISMUT , SHU
SHEN , AND ZHAOTING
WEI and let (cid:0)
E, A E (cid:48)(cid:48) (cid:1) = f ∗ b (cid:0) F, A F (cid:48)(cid:48) (cid:1) be the antiholomorphic superconnec-tion on X that was defined in Subsection 5.7.We fix a smooth splitting of F as in (5.2), (5.3), that induces a cor-responding smooth splitting of E on X . Then we have identifications F (cid:39) F , E (cid:39) E . Let h be a smooth section of M D F on Y . Then f ∗ h is a smooth section of M D E on Y . Proposition 8.5.
The following identity of forms on Y holds: (8.24) ch (cid:0) A f ∗ b E (cid:48)(cid:48) , f ∗ h (cid:1) = f ∗ ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) . Also (8.25) ch BC (cid:0) A f ∗ b E (cid:48)(cid:48) (cid:1) = f ∗ ch BC (cid:0) A E (cid:48)(cid:48) (cid:1) in H (=)BC ( Y, R ) . Proof.
The proof of equation (8.24) is easy and is left to the reader.By taking the corresponding Bott-Chern cohomology class in (8.24),we get (8.25). (cid:3)
Chern character and tensor products.
Let (cid:0)
E, A E (cid:48)(cid:48) (cid:1) , (cid:0) E, A E (cid:48)(cid:48) (cid:1) be antiholomorphic superconnections on X . Let (cid:16) E (cid:98) ⊗ b E, A E (cid:98) ⊗ b E (cid:48)(cid:48) (cid:17) de-note the corresponding tensor product, which was defined in Subsection5.8. Its diagonal bundle is given by D (cid:98) ⊗ D . We fix splittings for E, E similar to (5.2), (5.3), so that E (cid:39) E , E (cid:39) E . These two splittingsinduce a corresponding splitting of E (cid:98) ⊗ b E , and (cid:0) E (cid:98) ⊗ b E (cid:1) = E (cid:98) ⊗ b E .If h, h are smooth sections of M D , M D , then h (cid:98) ⊗ h is a smooth sectionof M D (cid:98) ⊗ D . Proposition 8.6.
The following identity of forms on X holds: (8.26) ch (cid:16) A ( E (cid:98) ⊗ b E ) (cid:48)(cid:48) , h (cid:98) ⊗ h (cid:17) = ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) . Also (8.27) ch BC (cid:16) A E (cid:98) ⊗ b E (cid:48)(cid:48) (cid:17) = ch BC (cid:0) A E (cid:48)(cid:48) (cid:1) ch BC (cid:0) A E (cid:48)(cid:48) (cid:1) in H (=)BC ( X, R ) . Proof.
The proof of (8.26) is easy and is left to the reader. Equation(8.27) is just a consequence. (cid:3)
The case where
H E is locally free.
We use the notation ofSubsection 5.3. We fix a splitting of E as in (5.2), (5.3). We write A E (cid:48)(cid:48) as in (5.6). Let h be a pure metric on D .In this Subsection, we assume that the O X -module H E = H HD is locally free. By (5.45), HD is a smooth holomorphic vector bundleon X .Put(8.28) H D = { f ∈ D, v f = 0 , v ∗ f = 0 } . OHERENT SHEAVES AND RRG 67
Then H is a smooth vector subbundle of D . Using finite-dimensionalHodge theory, we have a canonical isomorphism of smooth vector bun-dles,(8.29) H D (cid:39) HD.
As a subbundle of D , H D inherits a Hermitian metric. Let h HD denotethe corresponding smooth Hermitian metric on HD . Let ∇ HD be thecorresponding Chern connection on HD .Let P : D → H D denote the orthogonal projection from D on H D .Then P is a smooth section of End ( D ) that preserves the Z -grading.Let ∇ H D be the connection on H D ,(8.30) ∇ H D = P ∇ D . Then ∇ H D is a unitary connection on H D . Proposition 8.7.
Via the identification H D (cid:39) HD , (8.31) ∇ HD = ∇ HD . Proof.
Via the identification H D (cid:39) HD , we have the identity,(8.32) ∇ H D (cid:48)(cid:48) = ∇ HD (cid:48)(cid:48) . By (8.30), we get(8.33) ∇ H D (cid:48)(cid:48) = P ∇ D (cid:48)(cid:48) . By (8.33), and taking adjoints, we obtain(8.34) ∇ H D (cid:48) = P ∇ D (cid:48) . By (8.32)–(8.34), we get (8.31). The proof of our proposition is com-pleted. (cid:3)
The Chern character form and the scaling of the metric.
In this Subsection, we assume that h ∈ M D is a pure metric. Recallthat N D is the number operator of D , i.e., N D acts by multiplicationby k on D k . Definition 8.8.
For
T >
0, let h T ∈ M D be the pure metric,(8.35) h T = hT N D . Let A E (cid:48) T denote the adjoint of A E (cid:48)(cid:48) with respect to h T . Put(8.36) A E T = A E (cid:48)(cid:48) + A E (cid:48) T . BISMUT , SHU
SHEN , AND ZHAOTING
WEI If A E (cid:48) denote the adjoint of A E (cid:48)(cid:48) with respect to h , then(8.37) A E (cid:48) T = T − N D A E (cid:48) T N D . Set B E (cid:48)(cid:48) T = T N D / A E (cid:48)(cid:48) T − N D / ,B E (cid:48) T = T N D / A E (cid:48) T T − N D / , (8.38) B E T = T N D / A E T T − N D / . Then(8.39) B E T = B E (cid:48)(cid:48) T + B E (cid:48) T . Moreover, B E (cid:48) T is the h -adjoint of B E (cid:48)(cid:48) T .As we saw in Subsection 8.5, if H E is locally free, then HD is aholomorphic vector bundle, and the pure metric h induces a Hermitianmetric h HD on HD . Let ch (cid:0) HD, h HD (cid:1) be the Chern form associatedwith the Z -graded holomorphic Hermitian vector bundle HD . ThisChern form is defined using the methods of Subsection 2.4.Recall that the metric h det D on det D associated with h was definedafter equation (7.19). Let c (cid:0) det D, h det D (cid:1) denote the first Chern formassociated with the holomorphic Hermitian line bundle (cid:0) det D, h det D (cid:1) .The Knudsen-Mumford determinant line bundle det H E was con-sidered in Subsection 5.4, and we have the canonical isomorphism in(5.48).If α ∈ Ω (=) ( X, R ), we denote by α (1 , its component in Ω (1 , ( X, R ).If β ∈ H (=)BC ( X, R ), let β (1 , be the component of β in H (1 , ( X, R ).If α T | T > is a family of smooth forms on X , we will say that as T → + ∞ , α T = O (cid:16) / √ T (cid:17) if for any k ∈ N , there exists C k > T ≥
1, the sup of the norm of α T and its derivatives of order ≤ k is dominated by C k / √ T . Theorem 8.9. If H E is locally free, as T → + ∞ , (8.40) ch (cid:0) A E (cid:48)(cid:48) , h T (cid:1) = ch (cid:0) HD, h HD (cid:1) + O (cid:16) / √ T (cid:17) . Under the same assumption, (8.41) ch BC (cid:0) A E (cid:48)(cid:48) (cid:1) = ch BC ( HD ) in H (=)BC ( X, R ) . In particular, if HD = 0 , then (8.42) ch BC (cid:0) A E (cid:48)(cid:48) (cid:1) = 0 in H (=)BC ( X, R ) . As T → , (8.43) ch (1 , (cid:0) A E (cid:48)(cid:48) , h T (cid:1) → c (cid:0) det D, h det D (cid:1) . OHERENT SHEAVES AND RRG 69
We have the identity, (8.44) ch (1 , (cid:0) A E (cid:48)(cid:48) (cid:1) = c , BC (det H E ) in H (1 , ( X, R ) . Proof.
By (8.1), (8.38), we have(8.45) ch (cid:0) A E (cid:48)(cid:48) , h T (cid:1) = ϕ Tr s (cid:104) exp (cid:16) − B E , T (cid:17)(cid:105) . By (5.6), (7.15), and (8.38), we get B E (cid:48)(cid:48) T = √ T v + ∇ D (cid:48)(cid:48) + (cid:88) i ≥ T (1 − i ) / v i , (8.46) B E (cid:48) T = √ T v ∗ + ∇ D (cid:48) + (cid:88) i ≥ T (1 − i ) / v ∗ i . By (8.46), we get(8.47) B ET = √ T ( v + v ∗ ) + ∇ D + (cid:88) i ≥ T (1 − i ) / ( v i + v ∗ i ) . Using Proposition 8.7, (8.47), and proceeding as in [BGV92, Corol-lary 9.6] in a finite-dimensional context, we get (8.40). By (8.40), weobtain (8.41), and also (8.42).Set(8.48) B E , ≤ T = √ T ( v + v ∗ ) + ∇ D . Then B E T − B E , ≤ T is a smooth section of Λ ( T ∗ C X ) (cid:98) ⊗ End ( D ) whosedegree in Λ ( T ∗ C X ) is ≥
2. By proceeding as in (8.15), we deduce that(8.49) Tr s (cid:104) exp (cid:16) − B E , T (cid:17)(cid:105) (2) = Tr s (cid:104) exp (cid:16) − B E , ≤ , T (cid:17)(cid:105) (2) . As T → s (cid:104) exp (cid:16) − B E , ≤ , T (cid:17)(cid:105) (2) → Tr s (cid:2) exp (cid:0) −∇ D, (cid:1)(cid:3) (2) = Tr s (cid:2) −∇ D, (cid:3) . Using Proposition 7.5, we get(8.51) ϕ Tr s (cid:2) −∇ D, (cid:3) = c (cid:0) det D, h det D (cid:1) . By (8.49)–(8.51), we obtain (8.43). Finally, by combining Theorem 5.9and (8.43), we get (8.44). The proof of our theorem is completed. (cid:3)
Remark . The argument used in the proof of (8.49) is exactly thesame as the argument used by Bismut-Freed in a smooth infinite-dimensional context in their proof of [BF86, eq. (1.53) in Theorem1.19]. This argument reappears in a holomorphic infinite-dimensional
BISMUT , SHU
SHEN , AND ZHAOTING
WEI context in [BGS88c, Theorems 1.13 and 1.15]. However, in these ref-erences, the addition of ‘bad’ diverging terms in the considered super-connections plays a key role in the convergence of the superconnectionforms as T →
0, while here, the situation is the opposite. Observethat in [Q16, Theorem 4.21], when HD = 0, Qiang had establishedTheorem 8.9 by similar arguments to the ones we just gave.8.7. The Chern character of a cone.
We use the notation of Sub-sections 4.3 and 5.6. In particular, φ is a morphism E → E of degree0, and (cid:0) C, A C (cid:48)(cid:48) φ (cid:1) is the corresponding cone. Here we reobtain a resultof Qiang [Q16, Proposition 4.24]. Theorem 8.11.
The following identity holds: (8.52) ch BC (cid:0) A C (cid:48)(cid:48) φ (cid:1) = ch BC (cid:0) A E (cid:48)(cid:48) (cid:1) − ch BC (cid:0) A E (cid:48)(cid:48) (cid:1) in H (=)BC ( X, R ) . In particular, if φ is a quasi-isomorphism, then (8.53) ch BC (cid:0) A E (cid:48)(cid:48) (cid:1) = ch BC (cid:0) A E (cid:48)(cid:48) (cid:1) in H (=)BC ( X, R ) . Proof.
For t ∈ C , let (cid:0) C t , A C (cid:48)(cid:48) tφ (cid:1) be the complex of the cone associatedwith tφ as in Subsection 5.6. Recall that M t is given by (4.30). If t ∈ C ∗ , M t is invertible, and equation (5.68) holds. By Theorem 8.2,if t ∈ C ∗ ,(8.54) ch BC (cid:0) A C (cid:48)(cid:48) tφ (cid:1) = ch BC (cid:0) A C (cid:48)(cid:48) φ (cid:1) . Moreover, as t →
0, we get(8.55) ch BC (cid:0) A C (cid:48)(cid:48) tφ (cid:1) → ch BC (cid:0) A C (cid:48)(cid:48) (cid:1) . We have the trivial identity(8.56) ch BC (cid:0) A C (cid:48)(cid:48) (cid:1) = ch BC (cid:0) A E (cid:48)(cid:48) (cid:1) − ch BC (cid:0) A E (cid:48)(cid:48) (cid:1) . By (8.54)–(8.56), we get (8.52).If φ is a quasi-isomorphism, using equation (8.42) in Theorem 8.9,we get(8.57) ch BC (cid:0) A C (cid:48)(cid:48) φ (cid:1) = 0 . By combining (8.52) and (8.57), we get (8.53). The proof of our theo-rem is completed. (cid:3)
The Chern character on D bcoh ( X ) . By Theorem 6.7, if F isan object in D bcoh ( X ), there is an antiholomorphic superconnection E = (cid:0) E, A E (cid:48)(cid:48) (cid:1) such that E is isomorphic to F in D bcoh ( X ).Here we reobtain a result of Qiang [Q16, Theorem 4.25]. Theorem 8.12.
The class ch BC (cid:0) A E (cid:48)(cid:48) (cid:1) ∈ H (=)BC ( X, R ) depends only onthe isomorphism class of F in D bcoh ( X ) . OHERENT SHEAVES AND RRG 71
Proof.
Let F ∈ D bcoh ( X ) be isomorphic to F , and let E be a corre-sponding antiholomorphic superconnection. By Theorem 6.9, F X is anequivalence of categories. Therefore, there exists a quasi-isomorphism φ : E → E . Our theorem follows from Theorem 8.11. (cid:3) Definition 8.13.
Let ch BC ( F ) ∈ H (=)BC ( X, R ) be the common Bott-Chern class of the above ch BC (cid:0) A E (cid:48)(cid:48) (cid:1) .In the sequel, if E = (cid:0) E, A E (cid:48)(cid:48) (cid:1) , we will often identify E and F X ( E ),and we will also use the notation ch BC ( E ) instead of ch BC (cid:0) A E (cid:48)(cid:48) (cid:1) .Let X, Y be compact complex manifolds, and let f : X → Y bea holomorphic map. If F is an object in D bcoh ( Y ), then Lf ∗ F is anobject in D bcoh ( X ). Theorem 8.14.
The following identity holds: (8.58) ch BC ( Lf ∗ F ) = f ∗ ch BC ( F ) . Proof.
As before, we may and we will assume that instead E is anobject in B ( Y ). Our theorem is now a consequence of Propositions6.10 and 8.5. (cid:3) If F , F are objects in D bcoh ( X ), their derived tensor product F (cid:98) ⊗ L O X F was defined in Subsection 3.3. Theorem 8.15.
The following identity holds: (8.59) ch BC (cid:16) F (cid:98) ⊗ L O X F (cid:17) = ch BC ( F ) ch BC ( F ) in H (=)BC ( X, R ) . Proof.
We may as replace F , F by antiholomorphic superconnections E , E . Our Theorem follows from Propositions 6.11 and 8.6. (cid:3) The Chern character on K ( X ) . Let K (cid:0) D bcoh ( X ) (cid:1) denote the K -theory of D bcoh ( X ). By definition, this is the Grothendieck groupgenerated by the objects in D bcoh ( X ), with relations coming from theconstruction of cones. Namely if F , F ∈ D bcoh ( X ), if φ : F → F isa morphism of O X -complexes, let C ∈ D bcoh ( X ) be the cone of F , F .Then (8.60) C + F = F in K (cid:0) D bcoh ( X ) (cid:1) . Let K ( X ) be the Grothendieck group of coherent sheaves on X . By(3.2), (3.3), we have the identity(8.61) K (cid:0) D bcoh ( X ) (cid:1) = K ( X ) . This condition is equivalent to the standard condition using instead distin-guished triangles in D bcoh ( X ) [St20, Tag 0FCN]. BISMUT , SHU
SHEN , AND ZHAOTING
WEI
The derived tensor product on D bcoh ( X ) induces a corresponding tensorproduct on K ( X ), so that K ( X ) is a commutative ring.By Theorems 8.11 and 8.12, and by (8.61), the Chern character ch BC can be viewed as a morphism of rings from K ( X ) into H (=)BC ( X, R ).If F is an object in D bcoh ( X ), we will often identify F to its imagein K ( X ).Since the H i F are coherent sheaves, they can be viewed as elementsof D bcoh ( X ) equipped with the 0 differential. Put(8.62) RH F = (cid:88) ( − i H i F , so that RH F ∈ K ( X ). Theorem 8.16. If F ∈ D bcoh ( X ) , then (8.63) ch BC ( F ) = ch BC ( RH F ) . Proof.
By (3.2), (3.3), we have the identity,(8.64) F = RH F in K ( X ) . Taking the Chern character of (8.64), we get (8.63). The proof of ourtheorem is completed. (cid:3)
Remark . By Theorem 8.16, we deduce that if E = (cid:0) E, A E (cid:48)(cid:48) (cid:1) is anantiholomorphic superconnection, then(8.65) ch BC ( E ) = ch BC ( RH E ) . As we saw in Remark 5.8,
H E depends only on
D, v , ∇ D . By (8.65),the same is true for ch BC ( E ). When H E = H HD is locally free, thisis also a consequence of Theorem 8.9.If ∇ D (cid:48)(cid:48) defines a holomorphic structure on D , i.e., if ∇ D (cid:48)(cid:48) , = 0, wecan define ch BC ( D ). Proposition 8.18. If ∇ D (cid:48)(cid:48) , = 0 , then (8.66) ch BC ( E ) = ch BC ( D ) . Proof.
For t ∈ C , put(8.67) A E (cid:48)(cid:48) t = ∇ D (cid:48)(cid:48) + tv . Set(8.68) E t = (cid:0) E , A E (cid:48)(cid:48) t (cid:1) . By Remarks 5.8 and 8.17, for t (cid:54) = 0, H E = H E t . By (8.65), for t (cid:54) = 0,we get(8.69) ch BC ( E ) = ch BC ( E t ) . OHERENT SHEAVES AND RRG 73
Also as t ∈ C ∗ → BC ( E t ) → ch BC ( D ) . By (8.68), (8.70), we get (8.66). The proof of our proposition is com-pleted. (cid:3) If H E is locally free, our proposition is also a consequence of The-orem 8.9.8.10.
Spectral truncations.
We fix a splitting as in (5.2), (5.3), andalso a Hermitian metric g D on D . We write A E (cid:48)(cid:48) as in (5.6). Inparticular,(8.71) A E (cid:48)(cid:48) = v + A E (cid:48)(cid:48) ( ≥ . Put(8.72) A E = A E (cid:48)(cid:48) + v ∗ . As in (7.23), (7.24), we get A E , = (cid:2) A E , (cid:48)(cid:48) , v ∗ (cid:3) , (cid:2) A E (cid:48)(cid:48) , A E , (cid:3) = 0 , (cid:2) v ∗ , A E , (cid:3) = 0 . (8.73)Note that A E , is the piece of A E , in which the terms that containfactors of positive degree in Λ ( T ∗ X ) have been killed.Put(8.74) V = v + v ∗ . Then(8.75) V = [ v , v ∗ ] . Also Sp V ⊂ R + .Put(8.76) J = (cid:2) A E (cid:48)(cid:48) ( ≥ , v ∗ (cid:3) . Then J is a smooth section of degree 0 in Λ (cid:0) T ∗ X (cid:1) (cid:98) ⊗ End ( D ), whichonly contains terms of positive degrees in Λ (cid:0) T ∗ X (cid:1) .Then(8.77) A E , = V + J. Since Λ ( ≥ (cid:0) T ∗ X (cid:1) is nilpotent, from (8.77), we get(8.78) Sp A E , = Sp V . If λ ∈ C , λ / ∈ Sp ( V ), then(8.79) (cid:0) λ − A E , (cid:1) − = (cid:0) λ − V (cid:1) − + (cid:0) λ − V (cid:1) − J (cid:0) λ − V (cid:1) − + . . . and the expansion only contains a finite number of terms. BISMUT , SHU
SHEN , AND ZHAOTING
WEI
For a >
0, set(8.80) U a = (cid:8) x ∈ X, a / ∈ Sp V (cid:9) . Then U a is an open set in X . Definition 8.19.
Over U a , put(8.81) P a, − = 12 iπ (cid:90) λ ∈ C | λ | = a dλλ − A E , . Then P a, − is a projector that acts on E , and commutes with theaction of Λ (cid:0) T ∗ X (cid:1) . By (8.73), we get (cid:2) A E (cid:48)(cid:48) , P a, − (cid:3) = 0 , [ v ∗ , P a, − ] = 0 . (8.82)Put(8.83) P a, + = 1 − P a, − . Then P a, + is also a projector.By (8.79), (8.81), we can write P a, ± in the form(8.84) P a, ± = P a, ± + P ( ≥ a, ± . Let D a, ± be the direct sums of the eigenspaces of V for eigenvalues λ > a or λ < a . Then P a, ± is the orthogonal projectors D → D a, ± .Since P a, ± = P a, ± , we get(8.85) P ( ≥ a, ± = (cid:104) P a, ± , P ( ≥ a, ± (cid:105) + + P ( ≥ , a, ± . In particular P a, ± maps D a, ± into T ∗ X (cid:98) ⊗ D a, ∓ . Definition 8.20. On U a , put(8.86) E ,a, ± = P a, ± E . On U a , E ,a, ± is a subbundle of E which is also a Λ (cid:0) T ∗ X (cid:1) -module.Let i a, ± be the corresponding embedding in E . In the sequel, we usethe notation in Subsection 4.1, and in particular equation (4.2). Theorem 8.21. On U a , we have a splitting of Λ (cid:0) T ∗ X (cid:1) -modules, (8.87) E = E ,a, + ⊕ E ,a, − . Moreover, (8.88) F p E ,a, ± = E ,a, ± ∩ F p E . Also P a, ± induces a filtered isomorphism of Λ (cid:0) T ∗ X (cid:1) -modules, (8.89) Λ (cid:0) T ∗ X (cid:1) (cid:98) ⊗ D a, ± (cid:39) E ,a, ± . OHERENT SHEAVES AND RRG 75
As a Λ (cid:0) T ∗ X (cid:1) -module, E ,a, ± verifies the conditions in (4.6), and theassociated diagonal bundle is just D a, ± . Also it is equipped with a split-ting as in (5.2),(5.3).Moreover, A E (cid:48)(cid:48) preserves the smooth sections of E ,a, ± , and it in-duces an antiholomorphic superconnection A E ,a, ± (cid:48)(cid:48) on E ,a, ± , so thaton U a , we have the splitting (8.90) E = E ,a, + ⊕ E ,a, − . Finally, on U a , P a, − : E → E ,a, − and i a, − : E ,a, − → E are quasi-isomorphisms of O U a -complexes, and H E ,a, + = 0 .Proof. Equation (8.87) is obvious. Since P a, ± commutes with Λ (cid:0) T ∗ X (cid:1) ,it maps F p E into F p E ,a, ± , from which we get (8.88).Let Q a, ± be the morphism Λ p (cid:0) T ∗ X (cid:1) (cid:98) ⊗ D a, ± → F p E ,a, ± /F p +1 E ,a, ± induced by P a, ± . If e ∈ Λ p (cid:0) T ∗ X (cid:1) (cid:98) ⊗ D a, ± , and if P a, ± e ∈ F p +1 E ,a, ± , by(8.84), we find that e = 0, so that Q a, ± is injective. Let us prove Q a, ± is surjective. If e ∈ E ,a, ± , by (8.84), we deduce that(8.91) e = P a, ± e + P ( ≥ a, ± e, so that(8.92) e = P a, ± P a, ± e + P a, ± P ( ≥ a, ± e. If e ∈ F p E ,a, ± , by (8.92), e − P a, ± P a, ± e ∈ F p +1 E ,a, ± , which showsthat Q a, ± is surjective. By (8.89), we conclude that E ,a, ± verifies theconditions in (4.6), and it is equipped with a splitting as in (4.19) whichis induced by (8.89).By the first identity in (8.82), A E (cid:48)(cid:48) acts on smooth sections of E ,a, ± .Therefore, it induces an antiholomorphic superconnection A E ,a, ± (cid:48)(cid:48) on E ,a, ± , and (8.90) holds.Clearly, A E ,a, + , acts as an invertible operator on E ,a, + , v ∗ acts on E ,a, + , and by (8.73),(8.93) 1 | E ,a, + = (cid:104) A E ,a, + (cid:48)(cid:48) , v ∗ (cid:2) A E ,a, + , (cid:3) − (cid:105) . By (8.93), we find that
H E a, + = 0.By fiberwise Hodge theory, for any x ∈ X , P a, − : D → D a, − and i a, − : D a, − → D are quasi-isomorphisms. By Proposition 6.8, we concludethat P a, − and i a, − are quasi-isomorphisms.The proof of our theorem is completed. (cid:3) We will view A E ,a, ± (cid:48)(cid:48) as a superconnection A (cid:48)(cid:48) a, ± on Λ (cid:0) T ∗ X (cid:1) (cid:98) ⊗ D a, ± .By construction, we have the identity(8.94) P a, ± A (cid:48)(cid:48) a, ± = A E (cid:48)(cid:48) P a, ± . BISMUT , SHU
SHEN , AND ZHAOTING
WEI
Let us now express A (cid:48)(cid:48) a, ± as in (5.6), i.e.,(8.95) A (cid:48)(cid:48) a, ± = v ,a, ± + ∇ D a, ± (cid:48)(cid:48) + (cid:88) i ≥ v i,a, ± . By (8.84), (8.94), we can determine the various terms in (8.95) byrecursion.
Proposition 8.22.
The following identities hold: v ,a, ± = v | Da, ± , ∇ D a, ± (cid:48)(cid:48) = P a, ± ∇ D (cid:48)(cid:48) P a, ± . (8.96) Proof.
In degree 0, equation (8.94) gives the first equation in (8.96). Indegree 1, we get(8.97) ∇ D a, ± (cid:48)(cid:48) + P a, ± v ,a, ± = ∇ D (cid:48)(cid:48) P a, ± + v P a, ± . Since P a, ± exchanges D a, + and D a, − , from (8.97), we get the secondequation in (8.96). The proof of our proposition is completed. (cid:3) Remark . By Theorem 5.9, we know that ∇ D (cid:48)(cid:48) , ∇ D a, ± (cid:48)(cid:48) induce holo-morphic structures on the lines det D, det D a, ± . Since D = D a, + ⊕ D a, − ,we conclude that we have the smooth isomorphism(8.98) det D = det D a, + (cid:98) ⊗ det D a, − . By (8.97), it is elementary to verify that (8.98) is an isomorphismof holomorphic line bundles on U a . As explained in [BGS88a], since (cid:0) D a, + , v | D a, + (cid:1) is exact, det D a, + has a canonical section τ a, + . Since ∇ D a, + (cid:48)(cid:48) v ,a, + = 0, the section τ a, + is holomorphic. By the above, themap s ∈ det D a, − → s (cid:98) ⊗ τ a, + ∈ det D is a holomorphic identification ofline bundles.The above completely elucidates the arguments given in an infinitedimensional context in [BGS88c, Theorem 1.3], in which det D a, − wasshown directly to carry a holomorphic structure, The explanation isnow obvious: E a, − carries an antiholomorphic superconnection.Observe that for a < b , on U a ∩ U b , E ,a, − ⊂ E ,b, − , and that(8.99) P a, − = P a, − P b, − . In particular, on U a ∩ U b , P a, − | E ,b, − : E ,b, − → E ,a, − , and i a,b = E ,a, − → E ,b, − are quasi-isomorphisms.Let g D, be another Hermitian metric on D . We denote with anextra superscript 1 the objects we just constructed that are associatedwith g D, . Proposition 8.24.
For a > , a > , on U a ∩ U a , the map P a, − | E ,a , − : E ,a , − → E ,a, − is a quasi-isomorphism. OHERENT SHEAVES AND RRG 77
Proof.
Clearly,(8.100) P a, − | E ,a , − = P a, − i a , − . Our proposition follows from Theorem 8.21 and from (8.100). (cid:3) The case of embeddings
The purpose of this Section is to establish our Riemann-Roch-Grothendiecktheorem in the case of an embedding i X,Y : X → Y . Our proof usesall the properties we established in the previous Sections on ch BC , andalso the deformation to the normal cone. As a consequence we showthat our Chern character ch BC : K ( X ) → H (=)BC ( X, R ) verifies theuniqueness conditions stated by Grivaux [Gri10].This Section is organized as follows. In Subsection 9.1, we recallelementary facts on embeddings, direct images, and transversality.In Subsection 9.2, given an embedding i X,Y : X → Y , we describethe deformation to the normal cone.In Subsection 9.3, we establish our main theorem for the embedding i X,Y .Finally, in Subsection 9.4, from our Riemann-Roch-Grothendieck for-mula for embeddings, we show that our Chern character ch BC verifiesthe uniqueness conditions of Grivaux [Gri10].9.1. Embeddings, direct images, and transversality.
Let Z be acompact complex manifold. Let i X,Z : X → Z, i
Y,Z : Y → Z be twoholomorphic embeddings of compact complex manifolds. We assumethat U = Y ∩ Z is a compact submanifold of Z , and also that(9.1) T U = T X | U ∩ T Y | U . We denote by i U,X , i
U,Y the embeddings of U in X, Y .The excess normal bundle N on U is defined to be(9.2) N = T Z | U / ( T X | U + T Y | U ) . We have the exact sequence of vector bundles(9.3) 0 → N U/Y → N X/Z | U → N → . The manifolds X and Y are said to be transverse if N = 0, which isequivalent to(9.4) N U/Y = N X/Z | U . We use the notation of Section 3. Let F ∈ D bcoh ( X ), and let i X,Z, ∗ F ∈ D bcoh ( Z ) be its direct image, which coincides with the rightderived image Ri X,Z ∗ F . Let Li ∗ U,X F ∈ D bcoh ( U ) be the left derivedpull-back of F . Other similar objects will be denoted in the same way. BISMUT , SHU
SHEN , AND ZHAOTING
WEI
Proposition 9.1. If X and Y are transverse, there exists an isomor-phism in D bcoh ( Y ) , (9.5) Li ∗ Y,Z i X,Z, ∗ F (cid:39) i U,Y, ∗ Li ∗ U,X F . Proof.
Let E X = (cid:0) E X , A E X (cid:48)(cid:48) (cid:1) , E Z = (cid:0) E Z , A E Z (cid:48)(cid:48) (cid:1) be antiholomorphicsuperconnections on X, Z that represent F and i X,Z, ∗ F . By Propo-sition 6.10, Li ∗ U,X F is represented by i ∗ U,X,b E X , and Li ∗ Y,Z i X,Z, ∗ F isrepresented by i ∗ Y,Z,b E Z .As in Subsection 6.8, C ∞ ( X, E X ) can be viewed as a O ∞ Z (cid:0) Λ (cid:0) T ∗ Z (cid:1)(cid:1) -module, with the convention that if α ∈ Ω , • ( Z, C ) , s ∈ C ∞ ( X, E X ),then(9.6) α.s = (cid:0) i ∗ X,Z α (cid:1) s. We will denote the corresponding O Z -complex by i X,Z, ∗ E X . As we sawbefore, i X,Z, ∗ E X defines an object in D bcoh ( Z ), that coincides with itsright-derived direct image.By Proposition 6.12, there exists a morphism of O Z -complexes r Z,X : E Z → i X,Z, ∗ E X which induces also a morphism of O ∞ Z Λ (cid:0) T ∗ Z (cid:1) -modules,and r Z,X is a quasi-isomorphism of O Z -complexes.To make our notation simpler, we will use the notation E X insteadof i X,Z, ∗ E X . A similar notation will be used when considering the em-bedding i U,Y .The above maps can be combined in a commutative diagram (9.7) E Zi ∗ Y,Z,b (cid:15) (cid:15) r Z,X (cid:47) (cid:47) E Xi ∗ U,X,b (cid:15) (cid:15) i ∗ Y,Z,b E Z r
Y,U (cid:47) (cid:47) i ∗ U,X,b E X . in which r Y,U is induced by r Z,X . The first row consists of objects inD bcoh ( Z ), the second row of objects in D bcoh ( Y ). Also r Y,U is a morphismof O ∞ Y Λ (cid:0) T ∗ Y (cid:1) -modules, and of O Y -complexes.As we saw before, r Z,X is a quasi-isomorphism of O Z -complexes.To establish our proposition, we need to show that r Y,U is a quasi-isomorphism of objects in D bcoh ( Y ). The proof will be obtained vialocal arguments.Since r Z,X is a quasi-isomorphism, on Z \ X , H E Z = 0. By alocal version of Theorem 5.7, on Z \ X , HD Z = 0, so that on Y \ U , Hi ∗ Y,Z D Z = 0. Using again Theorem 5.7, we find that on Y \ U , H i ∗ Y,Z,b E Z = 0. As explained in Subsection 5.7, the subscript b is used to define the pull-backof antiholomorphic superconnections. OHERENT SHEAVES AND RRG 79
Take x ∈ U . If V ⊂ Z is a small neighborhood of x , we choosea holomorphic coordinate system on V such that x is represented by0 ∈ C n , and X, Y are two transverse vector subspaces H X , H Y of C n ,so that U is represented by H X ∩ H Y . If K is a vector subspace of H Y such that H X ⊕ K = C n , then K represents both N X/Z and N U/Y . Let V H X , V K be open neighborhoods of 0 in H X , K so that V H X × V K ⊂ V .By Theorem 5.3, if V H X is small enough, after a gauge transformationof total degree 0, A E X (cid:48)(cid:48) can be written in the form(9.8) A E X (cid:48)(cid:48) = ∇ D X (cid:48)(cid:48) + v X, . Let π H X , π K be the projections of C n on H X , K . Let y be the genericsection of K , and let (Λ K ∗ , i y ) denote the Koszul complex of K . Set(9.9) E Z = Λ (cid:0) C n (cid:1) (cid:98) ⊗ π ∗ H X D X (cid:98) ⊗ π ∗ K Λ K ∗ . Let A E Z (cid:48)(cid:48) be the antiholomorphic superconnection,(9.10) A E Z (cid:48)(cid:48) = π ∗ H X (cid:0) ∇ D X (cid:48)(cid:48) + v X, (cid:1) + π ∗ K (cid:16) ∂ K + i y (cid:17) . Then A E Z (cid:48)(cid:48) is an antiholomorphic superconnection on C n near 0. Let E Z denote the corresponding complex of O V -modules. Let r C n ,H X bethe projection π ∗ H X D X (cid:98) ⊗ π ∗ K Λ K ∗ → D X . Then r C n ,H X extends to amorphism E Z → E X which has the same properties as r Z,X . Becauseof known properties of the Koszul complex, if the considered neighbor-hoods are small enough, r C n ,H X is a quasi-isomorphism.Near z ∈ Z , both r Z,X and r C n ,H X provide quasi-isomorphisms of E Z , E Z with E X , so that for V small enough, E Z and E Z are isomorphicas objects in D bcoh ( V H X × V K ). By a local version of Theorem 6.9,there is a corresponding quasi-isomorphism φ : E Z → E Z . As we sawin Subsection 5.7, φ induces a morphism i ∗ Y,Z,b φ : i ∗ Y,Z,b E Z → i ∗ Y,Z,b E Z .By a local version of Proposition 6.8, for z ∈ V H X × V K , φ z : D Z,z → (cid:0) π ∗ H X D X (cid:98) ⊗ π ∗ K Λ K ∗ (cid:1) z is a quasi-isomorphism. In particular, this will betrue for z ∈ ( V H X × V K ) ∩ Y . This shows that near z ∈ Y , φ induces aquasi-isomorphism i ∗ Y,Z,b E Z → i ∗ Y,Z,b E Z .Since near x ∈ Y , the restriction of the above Koszul complex to Y is still a Koszul complex, r K,U : i ∗ Y,Z,b E Z → i ∗ U,X,b E X is a quasi-isomorphism. This shows that r Y,U : i ∗ Y,Z,b E Z → i ∗ U,X,b E X is a quasi-isomorphism. The proof of our proposition is completed. (cid:3) Deformation to the normal cone.
Let i X,Y : X → Y be aholomorphic embedding of compact complex manifolds. Let N X/Y bethe normal bundle to X in Y , i.e.,(9.11) N X/Y = i ∗ X,Y
T Y /T X.
BISMUT , SHU
SHEN , AND ZHAOTING
WEI
First, we construct the deformation of X to the normal cone to Y as in [BGS90b, Section 4]. Let W = W X/Y be the blow-up of Y × P along X × ∞ . Then X × P embeds in W . Let P be the exceptionaldivisor of the blow-up, i.e.,(9.12) P = P (cid:0) N X ×∞ /Y × P (cid:1) . Let p X , p ∞ be the projections X × ∞ → X, X × ∞ → ∞ . Put(9.13) A = p ∗ X N X/Y ⊗ p ∗∞ N − ∞ / P . Then(9.14) P = P ( A ⊕ C ) , so that P is the projective completion of A , and its divisor at ∞ isgiven by P ( A ) = P (cid:0) N X/Y (cid:1) .Let (cid:101) Y be the blow-up of Y along X . Then P (cid:0) N X/Y (cid:1) is the excep-tional divisor in (cid:101) Y . Let q W,Y : W → Y, q W, P : W → P be the obviousmaps. For z ∈ P , put(9.15) Y z = q − W, P z. Then Y z = Y if z (cid:54) = ∞ , (9.16) P ∪ (cid:101) Y if z = ∞ . For z = ∞ , P and (cid:101) Y meet transversally along P (cid:0) N X/Y (cid:1) . Also the pro-jection q W, P is a submersion except on P (cid:0) N Y/X (cid:1) where it has ordinarysingularities.Let U = O P ( −
1) be the universal line bundle on P . We have theexact sequence of holomorphic vector bundles on P ,(9.17) 0 → U → A ⊕ C → ( A ⊕ C ) /U → . The image σ of 1 ∈ C in ( A ⊕ C ) /U is a holomorphic section of( A ⊕ C ) /U that vanishes exactly on X × ∞ . On P ( A ), U restrictsto the corresponding universal line bundle, the exact sequence (9.17)restricts to(9.18) 0 → U → A ⊕ C → A/U ⊕ C → , and σ restricts to the section 1 of A/U ⊕ C .Consider the Koszul complex (Λ (( A + C ) /U ) ∗ , i σ ) on P . This com-plex provides a resolution of i X ×∞ ,P, ∗ O X ×∞ . On P (cid:0) N X/Y (cid:1) , this com-plex is just the split complex Λ (
A/U ) ∗ (cid:98) ⊗ (Λ C , i ). OHERENT SHEAVES AND RRG 81 (cid:102) Y P ( N X/Y ) P X × ∞ X × P X × Y Figure 1.
A Riemann-Roch-Grothendieck theorem for embeddings.
Consider the Bott-Chern class Td BC (cid:0) N X/Y (cid:1) ∈ H (=)BC ( X, R ).Let F ∈ D bcoh ( X ), and let i X,Y, ∗ F ∈ D bcoh ( Y ) be its direct image. Theorem 9.2.
The following identity holds: (9.19) ch BC ( i X,Y, ∗ F ) = i X,Y, ∗ ch BC ( F )Td BC (cid:0) N X/Y (cid:1) in H (=)BC ( Y, R ) . Proof.
Let i X × P ,W : X × P → W be the obvious embedding, let π X × P ,X : X × P → X be the natural projection. Then π ∗ X × P ,X F ∈ D bcoh ( X × P ) , so that(9.20) i X × P ,W, ∗ π ∗ X × P ,X F ∈ D bcoh ( W ) . Let i Y,W : Y = q − W, P (0) → W, i
P,W : P → W, (9.21) i (cid:101) Y ,W : (cid:101) Y → W, i X ×∞ ,P : X × ∞ → P. be the obvious embeddings. Other embeddings will be denoted in thesame way.Note that π X × P ,X i X,X × P is the identity of X , so that(9.22) Li ∗ X,X × P Lπ ∗ X × P ,X F (cid:39) F in D bcoh ( X ) . By [St20, Tag 00R4], π X × P ,X is flat, so that π ∗ X × P , X F = Lπ ∗ X × P ,X F . BISMUT , SHU
SHEN , AND ZHAOTING
WEI
Also π X × P ,X i X ×∞ ,X × P is the identification of X × ∞ with X . Whenidentifying these two spaces, we get(9.23) Li ∗ X ×∞ ,X × P Lπ ∗ X × P ,X F (cid:39) F in D bcoh ( X ) . Observe that Y = q − W, P (0) and P are transverse to X × P . UsingProposition 9.1, in which X, Y, Z, F are replaced by X × P , Y or P , W , Lπ ∗ X × P ,X F , and also (9.22), (9.23), we get (cid:0) Li ∗ Y,W (cid:1) i X × P ,W ∗ (cid:0) Lπ ∗ X × P ,X (cid:1) F (cid:39) i X,Y, ∗ F in D bcoh ( Y ) , (9.24) (cid:0) Li ∗ P,W (cid:1) i X × P ,W, ∗ (cid:0) Lπ ∗ X × P ,X (cid:1) F (cid:39) i X ×∞ ,P, ∗ F in D bcoh ( P ) . Using Theorem 8.14 and (9.24),we get i ∗ Y,W ch BC (cid:0) i X × P ,W, ∗ (cid:0) Lπ ∗ X × P ,X (cid:1) F (cid:1) = ch BC ( i X,Y, ∗ F ) in H (=)BC ( Y, R ) , (9.25) i ∗ P,W ch BC (cid:0) i X × P ,W, ∗ (cid:0) Lπ ∗ X × P ,X (cid:1) F (cid:1) = ch BC ( i X ×∞ ,P, ∗ F ) in H (=)BC ( P, R ) . Let α ∈ Ω (=) ( W, R ) be a smooth closed form on W representing(9.26) ch BC (cid:0) i X × P ,W, ∗ (cid:0) Lπ ∗ X × P ,X (cid:1) F (cid:1) ∈ H (=)BC ( W, R ) . By (9.25), i ∗ Y,W α ∈ Ω (=) ( Y, R ) represents ch BC ( i X,Y, ∗ F ) ∈ H (=)BC ( Y, R ),and i ∗ P,W α ∈ Ω (=) ( P, R ) represents ch BC ( i X ×∞ ,P, ∗ F ) ∈ H (=)BC ( P, R ).Let z ∈ C be the canonical meromorphic coordinate on P thatvanishes at 0. We have the Poincar´e-Lelong equation (2.25).Let δ Y , δ Y ∞ be the currents of integration on Y , Y ∞ . Since q W, P hasordinary singularities near P (cid:0) N X/Y (cid:1) , there is a well-defined integrablecurrent q ∗ W, P log (cid:0) | z | (cid:1) on W , which is such that(9.27) ∂ W ∂ W iπ q ∗ W, P log (cid:0) | z | (cid:1) = δ Y − δ Y ∞ . For z ∈ P , let i z be the embedding Y z → W . By (9.27), we deducethat(9.28) ∂ W ∂ W iπ (cid:2) αq ∗ W, P log (cid:0) | z | (cid:1)(cid:3) = i ∗ αδ Y − i ∗∞ αδ Y ∞ . Let q Y ∞ ,Y be the restriction of q W,Y to Y ∞ . By (9.28), we deduce theidentity of currents on Y ,(9.29) ∂ Y ∂ Y iπ q W,Y, ∗ (cid:2) αq ∗ W, P log (cid:0) | z | (cid:1)(cid:3) = i ∗ α − q Y ∞ ,Y, ∗ i ∗∞ α. OHERENT SHEAVES AND RRG 83
By the considerations in the paragraph following (9.25) and by (9.29),we get(9.30) ch BC ( i X,Y, ∗ F ) = { q Y ∞ ,Y, ∗ i ∗∞ α } in H (=)BC ( Y, R ) . Let q (cid:101) Y ,Y , q
P,Y be the restriction of q Y ∞ ,Y to (cid:101) Y , P . Note that(9.31) q Y ∞ ,Y, ∗ i ∗∞ α = q (cid:101) Y ,Y, ∗ i ∗ (cid:101) Y ,W α + q P,Y, ∗ i ∗ P,W α. Since (cid:101) Y ∩ ( X × ∞ ) = ∅ , we find that(9.32) Li ∗ (cid:101) Y ,W i X × P ,W, ∗ (cid:0) Lπ ∗ X × P ,X (cid:1) F (cid:39) bcoh (cid:16) (cid:101) Y (cid:17) . Using Theorem 8.14, the fact that { α } is given by (9.26), and (9.32),we get(9.33) (cid:110) i ∗ (cid:101) Y ,W α (cid:111) = 0 in H (=)BC (cid:16) (cid:101) Y , R (cid:17) , so that(9.34) (cid:110) q (cid:101) Y ,Y, ∗ i ∗ (cid:101) Y ,W α (cid:111) = 0 in H (=)BC ( Y, R ) . Using again the considerations in the paragraph following (9.25),(9.31), and (9.34), we obtain(9.35) ch BC ( i X,Y, ∗ F ) = q P,Y ∗ ch BC ( i X ×∞ ,P, ∗ F ) . We will now evaluate the right-hand side of (9.35).The Koszul complex ( O P (Λ (( A ⊕ C ) /U ) ∗ ) , i σ ) is a projective res-olution of the sheaf i X ×∞ ,P, ∗ O X ×∞ . By the projection formula [St20,Tag 0B54], we get(9.36) i X ×∞ ,P, ∗ F (cid:39) Lq ∗ P,X ×∞ F (cid:98) ⊗ L O P ( O P (Λ (( A ⊕ C ) /U ) ∗ ) , i σ ) in D bcoh ( P ) . By Theorems 8.14, 8.15 and by (9.36), we deduce that(9.37) ch BC ( i X ×∞ ,P, ∗ F ) = q ∗ P,X ×∞ ch BC ( F ) ch BC (Λ (( A ⊕ C ) /U ) ∗ ) . By (9.37), we get(9.38) q P,Y, ∗ ch BC ( i X ×∞ ,P, ∗ F ) = i X,Y, ∗ [ch BC ( F ) q P,X ×∞ , ∗ ch BC (Λ (( A ⊕ C /U ) ∗ ))] . Observe that(9.39) N ( X ×∞ ) /P = A. By [BGS90a, eq. (2.8) in Theorem 2.5] and by (9.39), we get(9.40) q P,X ×∞ , ∗ ch BC (Λ (( A ⊕ C ) /U ) ∗ ) = Td − ( A ) . BISMUT , SHU
SHEN , AND ZHAOTING
WEI
Recall that the choice of the coordinate z gives a canonical trivializationof N ∞ / P , so that by (9.13), we have the canonical identification(9.41) A (cid:39) N X/Y . By (9.35), (9.38), and (9.40), we get (9.19). The proof of our theoremis completed. (cid:3)
Remark . We claim that Proposition 9.1 and Theorem 9.2 are com-patible. Indeed, under the assumptions of Proposition 9.1, by Theorem9.2, we get(9.42) ch BC ( i X,Z, ∗ F ) = i X,Z, ∗ ch BC ( F )Td BC (cid:0) N X/Z (cid:1) in H (=)BC ( Z, R ) . If β ∈ Ω (=) ( X, R ) is a smooth form representing ch BC ( F )Td BC ( N X/Z ) , the right-hand side of (9.42) is represented by the current βδ X on Z .If X and Y are transverse, as explained in Subsection 2.3, i ∗ Y,Z βδ X is a well-defined current on Y , and it represents i ∗ Y,Z i X,Z, ∗ ch BC ( F )Td BC ( N X/Z ) in H (=)BC ( Y, R ). Moreover, one has the equality of currents on Y ,(9.43) i ∗ Y,Z ( βδ X ) = (cid:0) i ∗ U,X β (cid:1) δ U . By Theorem 8.14 and by (9.4), the Bott-Chern class of the currentin the right-hand side of (9.43) is just i U,Y, ∗ ch BC ( Li ∗ U,X F ) Td BC ( N U/Y ) . Using againTheorem 9.2 for the embedding i U,Y , we find that when taking theChern character of both sides of (9.5), we get a known equality.9.4.
The uniqueness of the Chern character.
First, we recall aversion of a result of Grivaux [Gri10, Theorem 2].
Theorem 9.4.
There is at most a unique morphism of groups ch BC : K ( X ) → H (=)BC ( X, R ) such that (1) If E is a holomorphic vector bundle, ch BC ( E ) is defined as inSubsection 2.4. (2) ch BC is functorial under pull-backs. (3) ch BC verifies Riemann-Roch-Grothendieck with respect to em-beddings.The Chern character ch BC constructed in Definition 8.13 verifies theabove uniqueness conditions.Proof. According to a modified version of Grivaux [Gri10], for unique-ness, we only need to check that if X is a compact complex manifold,if Y is a compact complex submanifold of X , and if σ : (cid:101) X → X is the OHERENT SHEAVES AND RRG 85 blow-up of X along Y , then σ ∗ : H (=)BC ( X, R ) → H (=)BC (cid:16) (cid:101) X, R (cid:17) is injec-tive. Since the exceptional divisor in (cid:101) X is negligible, if α ∈ Ω ( X, C ),(9.44) (cid:90) (cid:101) X σ ∗ α = (cid:90) X α. By (9.44), we deduce that if α ∈ Ω ( X, C ), we have the identity ofcurrents on X ,(9.45) σ ∗ σ ∗ α = α. From (9.45), we conclude that on H (=)BC ( X, R ), σ ∗ σ ∗ is the identity, andso σ ∗ is injective. This proves the uniqueness of ch BC .By construction, if E is a holomorphic vector bundle on X , ourch BC ( E ) is exactly the classical Chern character obtained in Subsec-tion 2.4. By Theorems 8.14 and 9.2, our ch BC verifies the above tworemaining conditions.The proof of our theorem is completed. (cid:3) Remark . The original unicity theorem of Grivaux would requirethat we prove if E is a holomorphic vector bundle on X , and if π : P → X is the total space of the projectivization P ( E ), then π ∗ : H (=)BC ( X, R ) → H (=)BC ( P, R ) is injective. We do not need to check this(which would be trivial here), because ch BC is known on all holomorphicvector bundles.If H (=)BC ( X, Q ) is the rational Bott-Chern cohomology [Sc07, Section4.e], Wu [W20] has constructed ch BC , Q : K ( X ) → H (=)BC ( X, Q ) a ra-tional Chern character. Let ι : H (=)BC ( X, Q ) → H (=)BC ( X, R ) be thecanonical map [Sc07, Section 7.a]. Corollary 9.6.
The following identity holds: ι ch BC , Q = ch BC . (9.46) Proof.
By [W20, Theorem 2], ι ch BC , Q satisfies the uniqueness condi-tions of Theorem 9.4, and so (9.46) holds. (cid:3) Submersions and elliptic superconnections
The purpose of this Section is to state our main result in the case ofa submersion p : M = X × S → S , and to define infinite dimensionalsuperconnection forms on S , which will be used to establish this mainresult.This Section is organized as follows. In Subsection 10.1, we state ourmain result, that involves F ∈ D bcoh ( M ). BISMUT , SHU
SHEN , AND ZHAOTING
WEI
In Subsection 10.2, using the results of Section 6, we show that inthe proof, F can be replaced by E ∈ B ( M ). We will then view p ∗ E as equipped with an antiholomorphic superconnection A p ∗ E (cid:48)(cid:48) .In Subsection 10.3, given a splitting E (cid:39) E , and Hermitian metrics g T X , g D on T X, D , we obtain the adjoint A p ∗ E (cid:48) of A p ∗ E (cid:48)(cid:48) .In Subsection 10.4, we construct natural connections on T R X .In Subsection 10.5, a Lichnerowicz formula is established for thecurvature of A p ∗ E , when S is reduced to a point.In Subsection 10.6, we give a Lichnerowicz formula for A p ∗ E , bytaking a proper adiabatic limit of the corresponding formula in the caseof a single fiber. This curvature is a second order elliptic differentialoperator along the fibers X .Finally, in Subsection 10.7, by imitating the constructions of Sec-tion 8 in an infinite-dimensional context, we obtain infinite-dimensionalChern character forms ch (cid:0) A p ∗ E (cid:48)(cid:48) , ω X , g D (cid:1) , and we show that they havethe same formal properties as the forms ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) that were consid-ered in Section 8. In particular, their class in Bott-Chern cohomologydoes not depend on the metric data.10.1. A theorem of Riemann-Roch-Grothendieck for submer-sions.
Let
X, S be compact complex manifolds of dimension n, n (cid:48) . Put(10.1) M = X × S. Then M has dimension m = n + n (cid:48) . Let p : M → S, q : M → X be theprojections. Then(10.2) T M = p ∗ T S ⊕ q ∗ T X.
Let F be an object in D bcoh ( M ). As we saw in Subsection 3.4, by atheorem by Grauert [Gr60, Theorem 10.4.6], Rp ∗ F ∈ D bcoh ( S ).The purpose of the next sections is to prove a special case of ourgeneral result. Theorem 10.1.
The following identity holds: (10.3) ch BC ( Rp ∗ F ) = p ∗ [ q ∗ Td BC ( T X ) ch BC ( F )] in H (=)BC ( S, R ) . Remark . We will exploit as much as possible the fact that M is product. However, the methods of [B13] can also be used to givea direct proof of Theorem 10.1 when p is an arbitrary holomorphicsubmersion.10.2. Replacing F by E . Let F ∈ D bcoh ( M ). By Theorem 6.7,there exists E = (cid:0) E, A E (cid:48)(cid:48) (cid:1) ∈ B ( M ) and a quasi-isomorphism of O X -complexes φ : E → F ∞ , so that H E = H F ∞ = H F . To prove ourTheorem, we may as well assume that F = E . OHERENT SHEAVES AND RRG 87
We will use the notation of Sections 5 and 7. In particular D denotesthe diagonal vector bundle on M which is associated with E .Note that p ∗ E is a Ω , • ( S, C )-module, and A E (cid:48)(cid:48) can be viewed as anantiholomorphic superconnection A p ∗ E (cid:48)(cid:48) on p ∗ E . The difference withrespect to Subsections 5.1–5.3 is that p ∗ E is infinite-dimensional.As we saw in Subsection 3.1, a theorem of Grauert [GrR84, Theorem10.4.6] asserts that Rp ∗ E is an object in D bcoh ( S ). Also by (3.11),(10.4) Rp ∗ E = p ∗ E . Therefore the O S -complex (cid:0) p ∗ E , A p ∗ E (cid:48)(cid:48) (cid:1) defines an object in D bcoh ( S ).This result will be given a direct proof in Theorem 11.6.Let i be the embedding of the fibers X into M . By Propositions5.11 and 6.10, the pull-back i ∗ b E = (cid:0) i ∗ b E, A i ∗ b E (cid:48)(cid:48) (cid:1) is a family of anti-holomorphic superconnections along the fibers X , whose correspondingassociated element in D bcoh ( X ) is just Li ∗ E .Let D be the diagonal bundle associated with p ∗ E . Then(10.5) D = p ∗ i ∗ b E , and the corresponding A D(cid:48)(cid:48) (which is an analogue of v for p ∗ E ) is givenby(10.6) A D(cid:48)(cid:48) = A i ∗ b E (cid:48)(cid:48) . The projection P defined in Subsection 5.1 is given here by i ∗ b : p ∗ E →D = p ∗ i ∗ b E .By (10.2), we get(10.7) Λ (cid:0) T ∗ M (cid:1) = p ∗ Λ (cid:0) T ∗ S (cid:1) (cid:98) ⊗ q ∗ Λ (cid:0) T ∗ X (cid:1) . By (5.1), (10.7), we obtain(10.8) E = p ∗ Λ (cid:0) T ∗ S (cid:1) (cid:98) ⊗ q ∗ Λ (cid:0) T ∗ X (cid:1) (cid:98) ⊗ D. Note that(10.9) i ∗ b E = q ∗ Λ (cid:0) T ∗ X (cid:1) (cid:98) ⊗ D. Over X × S , we fix splittings as in (5.2), (5.3), so that(10.10) E (cid:39) E . These splittings induce corresponding splittings of i ∗ b E , and we havecorresponding isomorphism,(10.11) i ∗ b E (cid:39) i ∗ b E . By (10.11), we get the non-canonical isomorphism(10.12) D = Ω , • ( X, D | X ) . BISMUT , SHU
SHEN , AND ZHAOTING
WEI
Put(10.13) p ∗ E = Λ (cid:0) T ∗ S (cid:1) (cid:98) ⊗ p ∗ i ∗ b E . By (10.5), (10.8), and (10.9), we have the identification ,(10.14) p ∗ E (cid:39) p ∗ E . To keep in line with the previous notation, we will denote A p ∗ E (cid:48)(cid:48) theoperator corresponding to A p ∗ E (cid:48)(cid:48) via the isomorphism (10.14).Using (5.9), we get(10.15) A p ∗ E (cid:48)(cid:48) = ∇ D (cid:48)(cid:48) + B. The adjoint of A p ∗ E (cid:48)(cid:48) . Since M = X × S , then ∇ D (cid:48)(cid:48) splits as(10.16) ∇ D (cid:48)(cid:48) = ∇ D (cid:48)(cid:48) ,X + ∇ D (cid:48)(cid:48) ,S , where the terms in the right-hand side of (10.16) differentiate along X, S .Let g T X be a Hermitian metric on
T X , and let ω X denote the cor-responding K¨ahler (1 , X . Let g T R X , g T ∗ R X bethe induced metrics on T R X, T ∗ R X . We denote by (cid:104) (cid:105) the correspond-ing scalar product on T R X . If J T R X denotes the complex structure of T R X , if U, V ∈ T R X , then (10.17) ω X ( U, V ) = (cid:10)
U, J T R X V (cid:11) . Let g Λ ( T ∗ X ) denote the metric induced by g T X on Λ (cid:0) T ∗ X (cid:1) . Let dx bethe volume form on X which is induced by g T X .Let g D be a Hermitian metric on the Z -graded vector bundle D .Then g D defines a pure metric h in M D .We equip Λ (cid:0) T ∗ X (cid:1) (cid:98) ⊗ D with the metric g Λ ( T ∗ X ) (cid:98) ⊗ g D . Definition 10.3. If s, s (cid:48) ∈ D = Ω , • ( X, D | X ), put(10.18) α ( s, s (cid:48) ) = (2 π ) − n (cid:90) X (cid:104) s, s (cid:48) (cid:105) g Λ ( T ∗ X ) ⊗ g D dx. It would be more appropriate here to introduce the sub-filtration of E associ-ated with p ∗ Λ (cid:0) T ∗ S (cid:1) . To simplify the presentation, we exploit as much as possible the product struc-ture of M = X × S , to avoid the difficulties with the more general situation con-sidered in [B13]. Here, we choose the sign conventions of Kobayashi-Nomizu [KN69, Chapter9.5] and of [BGS88b, BGS88c]. The opposite sign is also commonly used.
OHERENT SHEAVES AND RRG 89
Then α is a Hermitian product on D , that defines a pure metric in M D .Here we use the notation of Subsection 7.1, where the adjoint of anantiholomorphic superconnection with respect to a Hermitian metricwas defined. Definition 10.4.
Let A p ∗ E (cid:48) denote the adjoint of A p ∗ E (cid:48)(cid:48) with respectto α .Although A E (cid:48)(cid:48) and A p ∗ E (cid:48)(cid:48) are essentially the same object, their ad-joints A E (cid:48) and A p ∗ E (cid:48) are distinct. In particular, the metric g T X playsno role in the definition of A E (cid:48) . Still, we will show how to derive A p ∗ E (cid:48) from A E (cid:48) .If H is a smooth section of Λ (cid:0) T ∗ M (cid:1) (cid:98) ⊗ End ( D ), H ∗ still denotesthe adjoint of H associated with the metric h = g D , as defined inSubsection 4.4.Then H can be viewed as a smooth section ofΛ (cid:0) T ∗ S (cid:1) (cid:98) ⊗ End (cid:0) Λ (cid:0) T ∗ X (cid:1)(cid:1) (cid:98) ⊗ End ( D ) . Definition 10.5.
Let H ∗ α denote the adjoint of H with respect to themetric α on D .We use the conventions of Subsection 4.5, with V = T X, g V ∗ = g T ∗ X .Then c is a map Λ ( T ∗ C M ) → Λ ( T ∗ C S ) (cid:98) ⊗ R c ( T ∗ R X ). We extend c to amap Λ ( T ∗ C M ) (cid:98) ⊗ End ( D ) → Λ ( T ∗ C S ) (cid:98) ⊗ R c ( T ∗ R X ) (cid:98) ⊗ R End ( D ) . Also we use the identification in (4.43),(10.19) c ( T ∗ R X ) ⊗ R C = End (cid:0) Λ (cid:0) T ∗ X (cid:1)(cid:1) . Proposition 10.6. If H is a smooth section of Λ (cid:0) T ∗ M (cid:1) (cid:98) ⊗ End ( D ) ,then (10.20) H ∗ α = c H ∗ . Proof.
We may as well assume that D is trivial. We use the notationin (4.42). If f ∈ T ∗ X , f and i f ∗ are adjoint to each other with respectto the metric of Λ (cid:0) T ∗ X (cid:1) . From (4.42), we get (10.20). (cid:3) Recall that ∇ D (cid:48) was defined after (7.14). As in (10.16), we have thesplitting(10.21) ∇ D (cid:48) = ∇ D (cid:48) ,X + ∇ D (cid:48) ,S . We can extend ∇ D (cid:48) ,S to smooth sections of Λ ( T ∗ C S ) (cid:98) ⊗ Λ (cid:0) T ∗ X (cid:1) (cid:98) ⊗ D .Let ∇ D (cid:48)(cid:48) ,X ∗ α denote the formal adjoint of ∇ D (cid:48)(cid:48) ,X with respect to α .Then ∇ D (cid:48)(cid:48) ,X ∗ α is a classical Hermitian adjoint. BISMUT , SHU
SHEN , AND ZHAOTING
WEI
Proposition 10.7.
The following identity holds: (10.22) A p ∗ E (cid:48) = ∇ D (cid:48) ,S + ∇ D (cid:48)(cid:48) ,X ∗ α + c B ∗ . Proof.
This is a consequence of (10.15), (10.16), and of Proposition10.6. (cid:3)
Definition 10.8.
Put(10.23) A p ∗ E = A p ∗ E (cid:48)(cid:48) + A p ∗ E (cid:48) . Then A p ∗ E is a superconnection on p ∗ E .Put(10.24) ∇ D,S = ∇ D (cid:48)(cid:48) ,S + ∇ D (cid:48) ,S . By (7.21), (10.15), and (10.22), we get(10.25) A p ∗ E = ∇ D,S + ∇ D (cid:48)(cid:48) ,X + ∇ D (cid:48)(cid:48) ,X ∗ + c C. Let A D(cid:48)(cid:48)∗ denote the standard fiberwise L adjoint of A D(cid:48)(cid:48) . We usethe notation(10.26) A D = A D(cid:48)(cid:48) + A D(cid:48)(cid:48)∗ . Some connections on
T X . Put(10.27) β = 12 (cid:16) ∂ X − ∂ X (cid:17) iω X . Then β is a smooth section of Λ ( T ∗ R X ).Let ∇ T R X, LC be the Levi-Civita connection on T R X , and let ∇ T X bethe Chern connection on
T X . Let ∇ T R X be the connection on T R X which is induced by ∇ T X . The connection ∇ T R X, LC preserves T X ifand only if g T X is K¨ahler, and in this case, it coincides with ∇ T R X .Let ∇ Λ ( T ∗ X ) be the connection on Λ (cid:0) T ∗ X (cid:1) which is induced by ∇ T X . This connection preserves the Z -grading.Let ϑ be the canonical 1-form on X with values in T R X that corre-sponds to the identity. Let τ denote the torsion of ∇ T R X . Then τ isthe sum of a (2 , τ , with values in T X , and of a (0 , τ , with values in T X .Let (cid:104) τ ∧ , ϑ (cid:105) be the antisymmetrization of the tensor (cid:104) τ, ϑ (cid:105) . By [B89b,Proposition 2.1], or by (4.46), we get(10.28) (cid:104) τ ∧ , ϑ (cid:105) = − β. Also equation (10.28) determines τ . More precisely, ∂ X iω X = − (cid:10) τ , ∧ , ϑ , (cid:11) , ∂ X iω X = (cid:10) τ , ∧ , ϑ , (cid:11) . (10.29) OHERENT SHEAVES AND RRG 91
By (4.47), the connection ∇ T R X, − / on T R X associated with ∇ T X is such that if
U, V ∈ T R X , then(10.30) ∇ T R X, − / U V = ∇ T R XU V − τ ( U, V ) . As we saw at the end of Subsection 4.6, ∇ T R X, − / induces a connection ∇ T X, − / on T X , and the analogue of (4.59) holds.Here, we follow [B89b, Section 2 b)].
Definition 10.9.
Let ∇ T R X , ∇ T R X be the Euclidean connections on T R X whose torsions τ , τ are such that the tensors (cid:104) τ , ϑ (cid:105) , (cid:104) τ , ϑ (cid:105) aretotally antisymmetric, and moreover, (cid:104) τ ∧ , ϑ (cid:105) = 2 β, (cid:104) τ ∧ , ϑ (cid:105) = − β, (10.31)and let R T R X , R T R X be their curvatures.By [B89b, Proposition 2.5], the connection ∇ T R X preserves the com-plex structure, i.e., it induces a corresponding Hermitian connection ∇ T X on T X .Let A be the 1-form on X with values in antisymmetric sections ofEnd ( T R X ) such that if U, V, W ∈ T R X , then(10.32) (cid:104) A ( U ) V, W (cid:105) = − (cid:104) τ ( U, V ) , W (cid:105) + (cid:104) τ ( U, W ) , V (cid:105) . From the properties of τ given before, we find that for U ∈ T R X , A ( U )is a complex endomorphism of T R X . By [B90, Proposition 2.5 and eq.(2.37)], we have the identity(10.33) ∇ T X = ∇ T X + A. For t >
0, if ω X is replaced by ω X /t , β is replaced by β/t . However,the above connections are unchanged. Also if ω X is closed, all theconsidered connections on T R X coincide with ∇ T R X .By following [B89b, Section 2 b)] and [B13, Section 2.5], let us de-scribe the connection ∇ T R X in more detail. Put(10.34) F = T ∗ X ⊕ T X
Then F is a holomorphic Hermitian vector bundle on X . Let ∇ F = ∇ T ∗ X ⊕ ∇ T X denote the corresponding Chern connection.Let b be the smooth section of T ∗ X ⊗ Hom (
T X, T ∗ X ) such that if u, v ∈ T X ,then(10.35) b ( u ) v = i∂ X ω X ( u, v, · ) . Put(10.36) b = (cid:0) g T X (cid:1) − b ( g T X ) − . BISMUT , SHU
SHEN , AND ZHAOTING
WEI
Then b is a section of T ∗ X ⊗ Hom ( T ∗ X, T X ).Let ∇ F be the connection on F ,(10.37) ∇ F = (cid:20) ∇ T ∗ X b b ∇ T X (cid:21) . Then g T X identifies
T X with T ∗ X , so that(10.38) F (cid:39) T C X = T X ⊕ T X.
In [B89b, Theorem 2.9], [B13, Theorem 2.5.3], it was shown thatvia the identification F (cid:39) T C X , the connection ∇ F is real and itinduces the Euclidean connection ∇ T R X on T R X . If ∇ T C X denote thecorresponding connection on T C X , by (10.37), we get(10.39) ∇ T C X = ∇ T X (cid:16) g T X (cid:17) − b (cid:0) g T X (cid:1) − b ∇ T X . By (10.39), we have the confirmation that ∇ T R X is unchanged whenreplacing ω X by ω X /t .By (10.37), we get(10.40) ∇ F (cid:48)(cid:48) = (cid:20) ∇ T ∗ X (cid:48)(cid:48) b ∇ T X (cid:48)(cid:48) (cid:21) . If ∂ X ∂ X iω X = 0, by [B89b, Theorem 2.7], [B13, Theorem 2.5.3], ∇ F (cid:48)(cid:48) induces a new holomorphic structure on F . We have the exactsequence of holomorphic vector bundles,(10.41) 0 → T ∗ X → F → T X → . Equivalently, ∇ T C X (cid:48)(cid:48) defines a holomorphic structure on T C X , and theexact sequence (10.41) can be rewritten in the form,(10.42) 0 → (cid:16) T X, ∇ T X (cid:48)(cid:48) (cid:17) → (cid:0) T C X, ∇ T C X (cid:48)(cid:48) (cid:1) → (cid:0) T X, ∇ T X (cid:48)(cid:48) (cid:1) → . In (10.42), ∇ T X (cid:48)(cid:48) = ∇ T X (cid:48) depends on the metric g T X .If ∂ X ∂ X iω X = 0, then ∇ T C X (cid:48)(cid:48) is a holomorphic structure, so that R T R X is of type (1 , A, B, C, D ∈ T R X , then(10.43) (cid:68) R T R X ( A, B ) C, D (cid:69) = (cid:10) R T R X ( C, D ) A, B (cid:11) . Equation (10.43) is compatible with the fact that R T R X takes its valuesin complex endomorphisms of T R X . OHERENT SHEAVES AND RRG 93
Let ∇ Λ ( T ∗ X ) be the connection on Λ (cid:0) T ∗ X (cid:1) induced by the unitaryconnection ∇ T X on T X .Assume for the moment that X is spin, or equivalently that the linebundle det T X has a square root λ . Then λ is a holomorphic Hermitianline bundle. Let ∇ λ be the corresponding Chern connection. Put(10.44) S T R X = Λ (cid:0) T ∗ X (cid:1) ⊗ λ − . Then S T R X is the Hermitian vector bundle of (cid:0) T R X, g T R X (cid:1) -spinorsassociated with the underlying spin structure, the positive spinors cor-responding to even forms, and the negative spinors to odd forms.Also S T R X inherits a Hermitian connection ∇ S T R X , LC correspondingto ∇ T R X, LC , that preserves its Z -grading. Let ∇ Λ ( T ∗ X ) , LC denote theconnection on Λ (cid:0) T ∗ X (cid:1) that is induced by ∇ S T R X , ∇ λ . The connection ∇ Λ ( T ∗ X ) , LC preserves the Z -grading of Λ (cid:0) T ∗ X (cid:1) .The above considerations still make sense even if X is not spin, sincelocally, a square root λ always exists.By proceeding the way we did for ∇ T R X, LC , we define the connection ∇ Λ ( T ∗ X ) s to be the connection induced by ∇ T R X , ∇ λ on Λ (cid:0) T ∗ X (cid:1) .Let e , . . . , e n be an orthonormal basis of T R X , and let e , . . . , e n be the corresponding dual basis of T ∗ R X . Let c A be the section ofΛ ( T ∗ R X ) (cid:98) ⊗ c ( T R X ) given by(10.45) c A = 12 (cid:104) Ae i , e j (cid:105) c e ic e j . By equation (10.33), we get(10.46) ∇ Λ ( T ∗ X ) s = ∇ Λ ( T ∗ X ) + c A. Observe that i · β is a 1-form with values in Λ ( T ∗ R X ), so that c i · β isa 1-form with values in End (cid:0) Λ (cid:0) T ∗ X (cid:1)(cid:1) . By [B89b, eq. (2.34)] and by(10.28), we have the identity (10.47) ∇ Λ ( T ∗ X ) s = ∇ Λ ( T ∗ X ) , LC + c ( i · β ) . When ω X is replaced by ω X /t , β is replaced by β/t . However, c now dependson t . When suitably interpreted, (10.47) does not depend on the scaling. BISMUT , SHU
SHEN , AND ZHAOTING
WEI
In general, the connections ∇ Λ ( T ∗ X ) s and ∇ Λ ( T ∗ X ) do not coincide.More precisely, we have the identity (10.48) ∇ Λ ( T ∗ X ) = ∇ Λ ( T ∗ X ) s + 12 Tr T X [ A ] . Let ∇ Λ ( T ∗ X ) (cid:98) ⊗ D, LC be the connection on Λ (cid:0) T ∗ X (cid:1) (cid:98) ⊗ D that is in-duced by ∇ Λ ( T ∗ X ) , LC , ∇ D . Let ∇ Λ ( T ∗ X ) (cid:98) ⊗ D denote the connection onΛ (cid:0) T ∗ X (cid:1) (cid:98) ⊗ D which is associated with ∇ Λ ( T ∗ X ) , ∇ D .10.5. A Lichnerowicz formula for A D , . In this Subsection, we as-sume that S is reduced to a point. Let D X, LC denote the classical Diracoperator acting on C ∞ (cid:0) X, Λ (cid:0) T ∗ X (cid:1) (cid:98) ⊗ D (cid:1) . Namely, if e , . . . , e n is abasis T R X , and if e , . . . , e n is the corresponding dual basis of T ∗ R X ,then(10.49) D X, LC = c e i ∇ Λ ( T ∗ X ) (cid:98) ⊗ D, LC e i . Now we assume that e , . . . , e n is an orthonormal basis of T R X . Let K X be the scalar curvature of X , and let R D be the curvature of ∇ D .By Lichnerowicz formula, we get(10.50) D X, LC , = − ∇ Λ ( T ∗ X ) (cid:98) ⊗ D, LC , e i + 18 K X + c (cid:18) R D + 12 Tr (cid:2) R T X (cid:3)(cid:19) . In the right-hand side of (10.50), ∇ Λ ( T ∗ X ) (cid:98) ⊗ D, LC , e i is our notation forthe Bochner Laplacian. More precisely if e , . . . , e n is a locally definedorthonormal basis of T R X , then(10.51) ∇ Λ ( T ∗ X ) (cid:98) ⊗ D, LC , e i = (cid:88) ∇ Λ ( T ∗ X ) (cid:98) ⊗ D, LC , e i − ∇ Λ ( T ∗ X ) (cid:98) ⊗ D, LC (cid:80) ∇ T R X, LC ei e i Other Bochner Laplacians will be denoted in the same way, with thesame correction in the right-hand side. This point was clear in [B89b], and was only stated implicitly in [B12, eq.(3.72)]. Still observe that these two connections induce the same connection onEnd (cid:0) Λ (cid:0) T ∗ X (cid:1)(cid:1) . OHERENT SHEAVES AND RRG 95
Theorem 10.10.
The following identities hold: A D = D X, LC + c ( β + C ) ,A D , = − (cid:18) ∇ Λ ( T ∗ X ) (cid:98) ⊗ D, LC e i + c ( i e i ( β + C )) (cid:19) + K X c (cid:18) A E, + 12 Tr (cid:2) R T X (cid:3) − ∂ X ∂ X iω X (cid:19) + ( c ( β + C )) + 12 (cid:88) ( c ( i e i ( β + C ))) − c (cid:0) ( β + C ) (cid:1) , (10.52) ( c ( β + C )) + 12 (cid:88) ( c ( i e i ( β + C ))) − c (cid:0) ( β + C ) (cid:1) = (cid:88) i <... We use [B89b, Theorems 1.1, 1.3, and 2.2] and (10.25). Asexplained in Subsection 4.5, our normalization on the Clifford algebrais different from the one [B89b]. Also β + C is an odd section ofΛ ( T ∗ C X ) (cid:98) ⊗ End ( D ), while End ( D ) was absent in [B89b, Theorem 2.3].Still, for formal reasons, it is easy to check that the result of [B89b]remains correct. The proof of our theorem is completed. (cid:3) Remark . Observe that A D , = (cid:2) A D(cid:48)(cid:48) , A D(cid:48) (cid:3) . The standard conse-quences of Hodge theory hold true for A D , .By (10.47), (10.48), we get(10.53) − (cid:18) ∇ Λ ( T ∗ X ) (cid:98) ⊗ D, LC e i + c ( i e i ( β + C )) (cid:19) = − (cid:18) ∇ Λ ( T ∗ X ) (cid:98) ⊗ De i − 12 Tr T X [ A ( e i )] + c ( i e i C ) (cid:19) . Equation (10.53) makes clear that as it should be, the operator (10.53)is of total degree 0. Equations (10.52), (10.53) give a false sense ofsymmetry between the roles of β and C . In particular both c and β dodepend on g T X .10.6. A Lichnerowicz formula for A p ∗ E , . Since M = X × S → S is product, the connections on T X defined in Subsection 10.5 lift to q ∗ T X . Also the form β in (10.27) lifts to the form q ∗ β on M .Let A D , LC denote the Levi-Civita superconnection in the sense of[B86], [BGS88b, Section 2 a)],(10.54) A D , LC = ∇ D,S + D X, LC . BISMUT , SHU SHEN , AND ZHAOTING WEI Theorem 10.12. The following identities hold: A p ∗ E = A D , LC + c ( q ∗ β + C ) ,A p ∗ E , = − (cid:18) ∇ Λ ( T ∗ X ) (cid:98) ⊗ D, LC e i + c ( i e i ( q ∗ β + C )) (cid:19) + K X c (cid:18) A E , + 12 Tr (cid:2) R T X (cid:3) − q ∗ ∂ X ∂ X iω X (cid:19) + ( c ( q ∗ β + C )) + 12 (cid:88) ( c ( i e i ( q ∗ β + C ))) − c (cid:0) ( q ∗ β + C ) (cid:1) , (10.55) ( c ( q ∗ β + C )) + 12 (cid:88) ( c ( i e i ( q ∗ β + C ))) − c (cid:0) ( q ∗ β + C ) (cid:1) = (cid:88) i <... By (10.25), by the first identity in (10.52), and by (10.54), weget the first identity in (10.55).Let U be a small open set in S . We will view U as an open ball in C n (cid:48) , and we replace S × X by U × X . Given (cid:15) > 0, we equip T C n (cid:48) withthe constant metric g T C n (cid:48) which is K¨ahler, and T M with the metric p ∗ g T C n (cid:48) (cid:15) ⊕ q ∗ g T X . Let ω C n (cid:48) be the K¨ahler form of C n (cid:48) . If f , . . . , f n (cid:48) is an orthonormal basis of C n (cid:48) , and if f , . . . , f n (cid:48) is the correspondingdual basis, then(10.56) ω C n (cid:48) = − if α f α . Also √ (cid:15)f , . . . , √ (cid:15)f n (cid:48) is an orthonormal basis of T C n (cid:48) for the metric g T C n (cid:48) /(cid:15) .Let A D (cid:15) be the operator on U × X , which is analogue of the operator A D considered in Theorem 10.10. Since ω C n (cid:48) is closed, the analogue of β on U × X is just q ∗ β . Using Theorem 10.10, we get a Lichnerowiczformula for A D , (cid:15) . In this formula, c now depends on (cid:15) , and will insteadbe denoted c (cid:15) . Also the adjoint of f α is now (cid:15)i f α . We will take a properlimit of this formula as (cid:15) → A D (cid:15) = e − iω C n (cid:48) /(cid:15) A D (cid:15) e iω C n (cid:48) /(cid:15) . The effect of the conjugation is to change (cid:15)i f α into (cid:15)i f α − f α . In par-ticular, as (cid:15) → 0, after conjugation, we have the uniform convergence OHERENT SHEAVES AND RRG 97 together with all the derivatives,(10.58) c (cid:15) ( q ∗ β + C ) → c ( q ∗ β + C ) . By (10.49), by the first equation in (10.52), by (10.54), and by (10.58),we find that as (cid:15) → A D (cid:15) → A p ∗ E , in the sense that the coefficients of the considered differential operatorsand their derivatives of any order converge uniformly over compactsubsets.Now we take the limit as (cid:15) → A D , (cid:15) one derives from (10.52). It is now easy to take the limit as (cid:15) → A D , (cid:15) and to obtain (10.55). The proof of ourtheorem is completed. (cid:3) Remark . Since the fibration π : M → S is product, none of thesubtleties contained in [B86, BGS88b, B12] appears in our formula.10.7. The elliptic superconnection forms. By equation (10.55),the operator A p ∗ E , is elliptic along the fibers X , so that exp (cid:0) − A p ∗ E , (cid:1) is a fiberwise trace class operator.We will now imitate Definition 8.1 in an infinite dimensional context. Definition 10.14. Put(10.60) ch (cid:0) A p ∗ E (cid:48)(cid:48) , ω X , g D (cid:1) = ϕ Tr s (cid:2) exp (cid:0) − A p ∗ E , (cid:1)(cid:3) . Then ch (cid:0) A p ∗ E (cid:48)(cid:48) , ω X , g D (cid:1) is a smooth even form on S .Let P be the collection of parameters ω X , g D . Let d P be the deRham operator on P .The metric α in (10.18) depends on ω X , g D . Then α − d P α is a1-form with values in α -self-adjoint endomorphisms.Let w , . . . , w n be an orthonormal basis of T X with respect to g T X .Using (10.18), and proceeding as in [BGS88c, eq. (1.109) and Propo-sition 1.19], we get(10.61) α − d P α = − c d P iω X + 12 d P iω X ( w i , w i ) + (cid:0) g D (cid:1) − d P g D . Now we have the obvious analogue of Theorem 8.2. Theorem 10.15. The form ch (cid:0) A p ∗ E (cid:48)(cid:48) , ω X , g D (cid:1) lies in Ω (=) ( S, R ) , it isclosed, and its Bott-Chern cohomology class does not depend on ω X , g D ,or on the splitting of E . Also ϕ Tr s (cid:2) α − d P α exp (cid:0) A p ∗ E , (cid:1)(cid:3) is a -form BISMUT , SHU SHEN , AND ZHAOTING WEI on P with values in Ω (=) ( S, R ) , and moreover, (10.62) d P ch (cid:0) A p ∗ E (cid:48)(cid:48) , ω X , g D (cid:1) = − ∂ S ∂ S iπ ϕ Tr s (cid:2) α − d P α exp (cid:0) − A p ∗ E , (cid:1)(cid:3) . Proof. The proof of the fact that our forms are closed and lie in Ω (=) ( S, R )is formally the same as the proof of Theorem 8.2. By proceeding as inthe proof of Theorem 8.2, we get (10.62).Let us now prove that the Bott-Chern class of ch (cid:0) A p ∗ E (cid:48)(cid:48) , ω X , g D (cid:1) does not depend on the splitting of E . Let E (cid:48) = (cid:0) E (cid:48) , A E (cid:48) (cid:48)(cid:48) (cid:1) be anothersplitting of E . If M (cid:48) = M × P , and let p , p , p be the projections M (cid:48) → S, M (cid:48) → M, M (cid:48) → P . Consider E M (cid:48) = p ∗ ,b E , so that E M (cid:48) = (cid:0) E M (cid:48) , A E M (cid:48) (cid:48)(cid:48) (cid:1) . The diagonal bundle associated with E M (cid:48) is p ∗ D . Given z ∈ P , let j z denote the embedding M × { z } → M (cid:48) . For any z ∈ P ,then j ∗ z,b E M (cid:48) = E . Let E M (cid:48) , = (cid:0) E M (cid:48) , , A E M (cid:48) , (cid:48)(cid:48) (cid:1) be a splitting of E M (cid:48) such that j ∗ ,b E (cid:48) = E , and j ∗∞ ,b E M (cid:48) , = E (cid:48) . Let q be the projection M (cid:48) → S (cid:48) = S × P . We equip p ∗ D with the metric p ∗ g D . We will use our pre-vious results on the form ch (cid:0) A E M (cid:48) , (cid:48)(cid:48) , ω X , p ∗ g D (cid:1) . The pull-backs of thisform to S × S × ∞ are just ch (cid:0) A p ∗ E (cid:48)(cid:48) , ω X , g D (cid:1) , ch (cid:0) A p ∗ E (cid:48) (cid:48)(cid:48) , ω X , g D (cid:1) .Using the Poincar´e-Lelong equation (2.25), we get(10.63) ∂ S ∂ S iπ p , ∗ (cid:2) ch (cid:0) A q ∗ E M (cid:48) , (cid:48)(cid:48) , ω X , p ∗ g D (cid:1) p ∗ log (cid:0) | z | (cid:1)(cid:3) = ch (cid:0) A p ∗ E (cid:48)(cid:48) , ω X , g D (cid:1) − ch (cid:16) A p ∗ E (cid:48) (cid:48)(cid:48) , ω X , g D (cid:17) , from which we get the last statement in our theorem. (cid:3) Definition 10.16. Let ch BC (cid:0) A p ∗ E (cid:48)(cid:48) (cid:1) ∈ H (=)BC ( S, R ) be the commonBott-Chern cohomology class of the forms ch (cid:0) A p ∗ E (cid:48)(cid:48) , ω X , g D (cid:1) .11. Elliptic superconnection forms and direct images The purpose of this Section is to prove that, with the notation ofSection 10, ch BC (cid:0) A p ∗ E (cid:48)(cid:48) (cid:1) coincides with ch BC ( Rp ∗ E ).This Section is organized as follows. In Subsection 11.1, we applythe technique of spectral truncation of Subsection 8.10 to the infinite-dimensional p ∗ E . This will permit us to extend to p ∗ E results we hadestablished before only for ordinary antiholomorphic superconnections.In Subsection 11.2, we prove the critical result that there exists aclassical antiholomorphic superconnection E on S such that p ∗ E and E are quasi-isomorphic, the main point being that E is finite-dimensional.In Subsection 11.3, we prove our main identity. OHERENT SHEAVES AND RRG 99 Finally, in Subsection 11.4, we briefly consider the case where H Rp ∗ E is locally free, in which case our main result has a much simpler proof.We make the same assumptions, and we use the same notation as inSection 10.11.1. Spectral truncations in infinite dimensions. We will nowproceed in infinite dimensions as in Subsection 8.10. Recall that A D(cid:48)(cid:48)∗ is the standard L -adjoint of A D(cid:48)(cid:48) . Put(11.1) A p ∗ E = A p ∗ E (cid:48)(cid:48) + A D(cid:48)(cid:48)∗ . Equation (11.1) is the analogue of equation (8.72) for A E .As in (8.73), we get A p ∗ E , = (cid:2) A p ∗ E (cid:48)(cid:48) , A D(cid:48)(cid:48)∗ (cid:3) , (cid:2) A p ∗ E (cid:48)(cid:48) , A p ∗ E , (cid:3) = 0 , (cid:2) A D(cid:48)(cid:48)∗ , A p ∗ E , (cid:3) = 0 . (11.2)The operator A D , being a second order elliptic operator along thefibers X , it has discrete spectrum. As in (8.78), we get(11.3) Sp A p ∗ E , = Sp A D , . As in (8.80), for a > 0, set(11.4) U a = (cid:8) s ∈ S, a / ∈ Sp A D , (cid:9) . Then U a is open in S . Definition 11.1. Over U a , put(11.5) P a, − = 12 iπ (cid:90) λ ∈ C | λ | = a dλλ − A p ∗ E , . Then P a, − is a fiberwise smoothing projector acting on p ∗ E .By (8.73), we get (cid:2) A p ∗ E (cid:48)(cid:48) , P a, − (cid:3) = 0 , (cid:2) A D(cid:48)(cid:48)∗ , P a, − (cid:3) = 0 . (11.6)Put(11.7) P a, + = 1 − P a, − . Then P a, + is also a projector.As in (8.84), we get(11.8) P a, ± = P a, ± + P ( ≥ a, ± . Let D a, ± be the direct sum of the eigenspaces of A D , for eigenvalues λ > a or λ < a . Then P a, ± is the orthogonal projectors D → D a, ± .Also the analogue of (8.85) holds. Also D a, − is a finite-dimensionalvector bundle on U a . 00 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI Definition 11.2. Over U a , put(11.9) p ∗ E ,a, ± = P a, ± p ∗ E . Then p ∗ E ,a, ± is a subbundle of p ∗ E which is also a Λ (cid:0) T ∗ S (cid:1) -module.Let i a, ± be the corresponding embedding in p ∗ E .We use the notation of Subsection 4.1. We will establish an analogueof Theorem 8.21. The main difference is that here, p ∗ E is infinite-dimensional. Theorem 11.3. On U a , we have a splitting of Λ (cid:0) T ∗ S (cid:1) -modules, (11.10) p ∗ E = p ∗ E ,a, + ⊕ p ∗ E ,a, − . Moreover, (11.11) F p p ∗ E ,a, ± = p ∗ E ,a, ± ∩ F p p ∗ E . Also P a, ± induces a filtered isomorphism of Λ (cid:0) T ∗ S (cid:1) -modules, (11.12) Λ (cid:0) T ∗ S (cid:1) (cid:98) ⊗D a, ± (cid:39) p ∗ E ,a, ± . As a Λ (cid:0) T ∗ S (cid:1) -module, p ∗ E ,a, − verifies the conditions in (4.6), and theassociated diagonal bundle is just D a, − . Also it is equipped with a split-ting as in (5.2), (5.3).Moreover, A p ∗ E (cid:48)(cid:48) preserves the smooth sections of p ∗ E ,a, ± , and itinduces an antiholomorphic superconnection A p ∗ E ,a, ± (cid:48)(cid:48) on p ∗ E ,a, ± .On U a , P a, − : E → E ,a, − and i a, − = E ,a, − → E are quasi-isomorphisms of O U a -complexes, and H E ,a, + = 0 . Finally, for s ∈ U a ,the complex D a, + ,s is exact.Proof. The proof is essentially the same as the proof of Theorem 8.21.With respect to that Theorem, here, only D a, − is finite-dimensional.Let us now prove that if s ∈ U a , D a, + is exact. By proceeding as inthe proof of Proposition 8.22, i.e., using the analogue of (8.94), thepart of degree 0 over Λ (cid:0) T ∗ S (cid:1) of A p ∗ E ,a, + is just the restriction of A D(cid:48)(cid:48) to D a, + . By Hodge theory, the complex D a, + ,s is exact. The proof of ourtheorem is completed. (cid:3) When considering another pair of metrics g T X , g D on T X, D , theobvious analogue of Proposition 8.24 still holds. Details are left to thereader.As a consequence of Theorem 11.3, we give another proof of a theo-rem of Grauert [GrR84, Theorem 10.4.6]. Theorem 11.4. The complex of O S -modules p ∗ E defines an elementin D bcoh ( S ) . Moreover, the Leray spectral sequence associated with thefiltration of p ∗ E by Λ (cid:0) T ∗ S (cid:1) degenerates at E . OHERENT SHEAVES AND RRG 101 Proof. By Theorem 11.3, over U a , p ∗ E and p ∗ E ,a, − are quasi-isomorphic.By Theorem 5.7, H p ∗ E ,a, − is a coherent sheaf. This shows that p ∗ E defines an object in D bcoh ( S ). By Theorem 5.7, the spectral sequenceassociated with p ∗ E ,a, − degenerates at E . Also on D a, + , we have theidentity(11.13) 1 = (cid:104) A D(cid:48)(cid:48) , A D(cid:48)(cid:48)∗ (cid:2) A D , (cid:3) − (cid:105) . Equation (11.13) shows that on p ∗ E ,a, + , the spectral sequence stops at E . This concludes the proof of our theorem. (cid:3) Remark . In the context of the theory of determinant bundles ofdirect images, Theorems 11.3 and 11.4 can be used as a substitute ofmany arguments in [BGS88c, Section 3], where non-projective K¨ahlermanifolds are also considered.11.2. The existence theorem. Recall that (10.13), (10.14) hold, andthat A p ∗ E (cid:48)(cid:48) is given by (10.15).Recall that if F ∈ D bcoh ( M ), RH F ∈ K ( M ) was defined in (8.62).As we saw in Theorem 11.4, p ∗ E is an object in D bcoh ( S ). Theorem 11.6. There exists a finite-dimensional antiholomorphic su-perconnection E = (cid:0) E, A E (cid:48)(cid:48) (cid:1) on S , and a morphism of O ∞ S (cid:0) Λ (cid:0) T ∗ S (cid:1)(cid:1) -modules φ : E → p ∗ E which is a quasi-isomorphism of O S -complexes,such that for any s ∈ S , φ s : ( D, v ) s → (cid:0) D , A D(cid:48)(cid:48) (cid:1) s is a quasi-isomorphism.We have the identity (11.14) E (cid:39) Rp ∗ E in D bcoh ( S ) . In particular, we have the identity, (11.15) RH E = p ! [ RH E ] in K ( S ) . Proof. By Proposition 6.12, there exists an object E in B ( S ) and amorphism of O ∞ S (cid:0) Λ (cid:0) T ∗ S (cid:1)(cid:1) -modules φ : E → p ∗ E , which is also aquasi-isomorphism of O S -complexes. Using Theorem 11.3, over U a , if φ a = P a, − φ , φ a induces a quasi-isomorphism E → p ∗ E ,a, − .Over U a , we now deal with finite dimensional antiholomorphic su-perconnections. By Proposition 6.8, for s ∈ U a , φ a induces a quasi-isomorphism D s → D a, − ,s . By Theorem 11.3, for s ∈ U a , the complex D a, + ,s is exact. Therefore, for any s ∈ U a , φ s : D s → D s is a quasi-isomorphism.Since φ is a quasi-isomorphism, we get (11.14), from which (11.15)follows. The proof of our theorem is completed. (cid:3) 02 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI Remark . By equation (11.14) in Theorem 11.6, if E (cid:48) has the sameproperties as E , then(11.16) E (cid:39) E (cid:48) in D bcoh ( S ) . Using Theorem 6.9 and (11.16), we deduce that there exists a quasi-isomorphism ψ : E → E (cid:48) . By Proposition 6.8, this just says that if D, D (cid:48) are the corresponding diagonal bundles on S , for any s ∈ S , ψ s : D s → D (cid:48) s is a quasi-isomorphism.11.3. The Chern character of the direct image. We use the no-tation of Theorem 11.6. In particular E is taken as in this Theorem. Theorem 11.8. The following identity holds: (11.17) ch BC (cid:0) A p ∗ E (cid:48)(cid:48) (cid:1) = ch BC ( E ) in H (=)BC ( S, R ) . Proof. The proof is an adaptation of the proof of Theorems 8.9 and8.11. We will use the notation in Theorem 11.6. For t ∈ C , we formthe cone C t = cone ( E , p ∗ E ) associated with the morphism tφ , withunderlying infinite-dimensional vector bundle C t on S . Note that φ induces a morphism D → D . The diagonal bundle D associated with C t is given D t = cone ( D, D ), where the morphism is induced by tφ .Also(11.18) D • t = D • +1 ⊕ D • . By Theorem 11.6, for any s ∈ S , when t (cid:54) = 0, D t is exact.We fix splittings of E, E as in (5.2), (5.3). In the sequel, we mayas well replace E , E by E , E . In particular C t also splits and can bereplaced by C t, .Let g D be a Hermitian metric on D . Let g D denote the direct sumHermitian metric on D associated with g D , g D . We equip C t, withthe pure metric g D . Let (cid:3) D t denote the corresponding fiberwise HodgeLaplacian acting on D . Since for any s ∈ S , D t is exact, (cid:3) D t,s isinvertible, and it has a uniform positive lower bound over S . Its heatkernel is again fiberwise trace class.Let A C t, (cid:48)(cid:48) denote the antiholomorphic superconnection on C t that isassociated with A E (cid:48)(cid:48) , A p ∗ E (cid:48)(cid:48) , tφ .Given t ∈ C , we can still define the Chern form ch (cid:0) A C t, (cid:48)(cid:48) , ω X , g D , g D (cid:1) ,for which the analogue of Theorem 10.15 holds, with a similar proof.Let ch BC (cid:0) A C t, (cid:48)(cid:48) (cid:1) ∈ H (=)BC ( S, R ) denote the corresponding commonBott-Chern cohomology class.For T > 0, let g DT , g DT be the metrics on D, D constructed as in(8.35). Let g D T be the metric on D that is associated with the metrics OHERENT SHEAVES AND RRG 103 ω X /T, g DT , g DT T − n − . If N D is the number operator of D , then(11.19) g D T = T − n g D T N D . The proof of (11.19) is elementary. The extra factor T − n comes fromthe contribution of the volume form on X to the Hermitian product on D . This normalizing factor does not contribute to the computation ofthe adjoint A C t, (cid:48) T of A C t, (cid:48)(cid:48) . By (11.19), if (cid:3) D t,T denotes the associatedfiberwise Laplacian, then(11.20) (cid:3) D t,T = T (cid:3) D t . We claim that if t (cid:54) = 0, as T → + ∞ ,(11.21) ch (cid:16) A C t, (cid:48)(cid:48) , ω X /T, g DT , g DT T − n − (cid:17) → . The proof is an infinite-dimensional analogue of equation (8.40) in The-orem 8.9, with HD = 0 which can be established by the methods of[BGV92, Theorems 9.19 and 9.23], [B97, Theorems 9.5 and 9.6].By (11.21), we deduce that for t (cid:54) = 0,(11.22) ch BC (cid:0) A C t, (cid:48)(cid:48) (cid:1) = 0 . By taking t → BC (cid:0) A C , (cid:48)(cid:48) (cid:1) = 0 . As in (8.56), we have the identity(11.24) ch BC (cid:0) A C , (cid:48)(cid:48) (cid:1) = ch BC (cid:0) A p ∗ E (cid:48)(cid:48) (cid:1) − ch BC ( E ) . By (11.23), (11.24), we get (11.17). The proof of our theorem is com-pleted. (cid:3) Remark . By Theorem 8.12 and Remark 11.7, we already knowthat ch BC ( E ) does not depend on the choice of E . Theorem 11.8 givesa direct proof of this fact. Theorem 11.10. The following identity holds: (11.25) ch BC (cid:0) A p ∗ E (cid:48)(cid:48) (cid:1) = ch BC ( Rp ∗ E ) in H (=)BC ( S, R ) . Proof. Using equation (11.14) in Theorem 11.6 and Theorem 11.8, weget (11.25). (cid:3) 04 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI The case where H Rp ∗ E is locally free. By Theorem 8.16,we know that(11.26) ch BC ( E ) = ch BC ( RH E ) in H (=)BC ( S, R ) . Also H Rp ∗ E is a Z -graded coherent sheaf on S . Using equation (11.17)in Theorem 11.10 and (11.26), we get(11.27) ch BC (cid:0) A p ∗ E (cid:48)(cid:48) (cid:1) = ch BC ( H Rp ∗ E ) in H (=)BC ( S, R ) . When H Rp ∗ E is locally free, there is a simpler proof of Theorem11.10 not relying on Theorem 11.6. In this case, the same arguments asin Subsection 8.5 show that the cohomology of the complex (cid:0) D , A D(cid:48)(cid:48) (cid:1) is a finite-dimensional holomorphic vector bundle H on S . Using fiber-wise Hodge theory, we can identify H with the corresponding fiberwiseharmonic sections H . Let g H denote the corresponding Hermitian met-ric on H .Under the above assumption, a strict analogue of Theorem 8.9 canbe proved. More precisely, by proceeding as in Berline-Getzler-Vergne[BGV92, Theorems 9.19 and 9.23], [B97, Theorems 9.5 and 9.6], andusing the obvious analogue of (11.19), one can show that as T → + ∞ ,we have the uniform convergence of forms on S ,(11.28) ch (cid:0) A p ∗ E (cid:48)(cid:48) , ω X /T, g DT (cid:1) → ch (cid:0) H, g H (cid:1) . By (11.28), we deduce that(11.29) ch BC (cid:0) A p ∗ E (cid:48)(cid:48) (cid:1) = ch BC ( H ) in H (=)BC ( S, R ) . Moreover,(11.30) H Rp ∗ E = H in K ( S ) . Using (11.29), (11.30), we reobtain Theorem 11.10.The above strategy was used in Bismut [B13, Theorems 4.10.4 and4.11.2] in the case where D is a holomorphic vector bundle on M , and (cid:0) E, A E (cid:48)(cid:48) (cid:1) = (cid:0) D, ∇ D (cid:48)(cid:48) (cid:1) , in order to establish Theorem 11.10.12. A proof of Theorem 10.1 when ∂ X ∂ X ω X = 0 . We make the same assumptions as in Section 10. We will prove The-orem 10.1 under the assumption that ∂ X ∂ X ω X = 0, by extending [B13,Theorem 5.1.2]. Under this assumption, by the results of Subsections2.4 and 10.4, R T R X is a (1 , 1) form, so that (cid:98) A (cid:0) T R X, ∇ T R X (cid:1) is a closedform in Ω (=) ( X, R ). OHERENT SHEAVES AND RRG 105 Theorem 12.1. If ∂ X ∂ X ω X = 0 , as t → , we have the uniformconvergence of forms on S , (12.1) ch (cid:0) A p ∗ E (cid:48)(cid:48) , ω X /t, g D (cid:1) → p ∗ (cid:104) q ∗ (cid:104) (cid:98) A (cid:0) T R X, ∇ T R X (cid:1) e c ( T X,g TX ) / (cid:105) ch (cid:0) A E (cid:48)(cid:48) , g D (cid:1)(cid:105) . In particular, if ω X is closed, as t → , (12.2) ch (cid:0) A p ∗ E (cid:48)(cid:48) , ω X /t, g D (cid:1) → p ∗ (cid:2) q ∗ Td (cid:0) T X, g T X (cid:1) ch (cid:0) A E (cid:48)(cid:48) , g D (cid:1)(cid:3) . If ∂ X ∂ X ω X = 0 , then (12.3) ch BC ( Rp ∗ E ) = p ∗ [ q ∗ Td BC ( T X ) ch BC ( E )] in H (=)BC ( S, R ) . Proof. In the proof, we will assume that ∂ X ∂ X ω X = 0. Also, we fix s ∈ S . The analysis will be done in the fiber X × { s } .Given t > , x, x (cid:48) ∈ X , let P t ( x, x (cid:48) ) denote the smooth kernel forexp (cid:16) − A p ∗ E , t (cid:17) along the fiber X × { s } with respect to the volume dx (cid:48) / (2 π ) n . Then(12.4) ch (cid:0) A p ∗ E (cid:48)(cid:48) , ω X /t, g D (cid:1) = ϕ (cid:90) X Tr s [ P t ( x, x )] dx (2 π ) n . To calculate the asymptotics of (12.4) as t → 0, we will use the localindex theoretic methods of [B86, BGS88c, B12].We denote by c t the map c associated with the metric g T X /t on T X ,that induces the metric tg T ∗ X on T ∗ X . By (4.42), if f ∈ T ∗ X , if f ∗ ∈ T X corresponds to f by the metric g T X , then c t (cid:0) f (cid:1) = f ∧ , c t ( f ) = − i tf ∗ . (12.5)Let A p ∗ E t denote the superconnection A p ∗ E associated with ω X /t, g D .In the right-hand side of the equation for A p ∗ E , t in (10.55), by (10.53),in the first row of the right-hand side, the first term is now(12.6) − t (cid:18) ∇ Λ ( T ∗ X ) ⊗ De i − 12 Tr T X [ A ( e i )] + c t ( i e i C ) (cid:19) . The second term is also scaled by the factor t . The term containing ∂ X ∂ X iω X is absent in the right-hand side. Also the e i are replaced by √ te i .We will compute the limit as t → s [ P t ( x, x )]. We fix x ∈ X . We take geodesic coordinates on X near x with respect to themetric g T X . This way, for (cid:15) > x ∈ X , weidentify the open ball B T R ,x X (0 , (cid:15) ) in T R ,x X to a corresponding open In the proof, the parameter s will be usually omitted. 06 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI geodesic ball B X ( x, (cid:15) ) in X . Given Y ∈ T R ,x X , along the geodesic u ∈ R → exp x ( uY ) ∈ X , we trivialize T X, D by parallel transportwith respect to the connections ∇ T X , ∇ D . Also we trivialize Λ (cid:0) T ∗ X (cid:1) via the connection ∇ Λ ( T ∗ X ). If w , . . . , w n is an orthonormal basis of T x X , it can also be viewed as an orthonormal trivialization of T X near x . In this trivialization, we have the identity(12.7) ω X = − iw i w i . If e , . . . , e n is an orthonormal basis of T R ,x X , via the parallel trans-port with respect to ∇ T R X along the above geodesics, we obtain anorthonormal basis of T R X near x .Using the above trivialization, near x , the operator A p ∗ E , t acts on H x,(cid:15) = C ∞ (cid:0) B T R ,x (0 ,(cid:15) ) , Λ (cid:0) T ∗ C ,s S (cid:1) (cid:98) ⊗ Λ (cid:0) T ∗ x X (cid:1) (cid:98) ⊗ D s,x (cid:1) . A first standard step is to construct an elliptic operator acting on H x = C ∞ (cid:0) T R ,x X, Λ (cid:0) T ∗ C ,s S (cid:1) (cid:98) ⊗ Λ (cid:0) T ∗ x X (cid:1) (cid:98) ⊗ D s,x (cid:1) , that coincides with A p ∗ E , t on B T R ,x X (0 , (cid:15)/ s [ P t ( x, x )] as t → A p ∗ E , t acts on H x , and that P t ( Y, Y (cid:48) ) is now a kernel on T R ,x X .For a > , u ∈ H x , put(12.8) F a u ( Y ) = u ( aY ) . Set(12.9) L t,x = F √ t A p ∗ E , t F / √ t . Let Q t,x ( Y, Y (cid:48) ) , Y, Y (cid:48) ∈ T R ,x X be the smooth kernel for exp ( − L t,x )with respect to the volume form dY (cid:48) (2 π ) n . Clearly(12.10) P t ( x, x ) = t − n Q t,x (0 , . Among the monomials in the w i , i w j , observe that up to obvious per-mutations, the only monomial whose supertrace on Λ (cid:0) T ∗ x X (cid:1) is nonzerois (cid:81) ni =1 w i i w i , and moreover(12.11) Tr sΛ ( T ∗ x X ) (cid:34) n (cid:89) i =1 w i i w i (cid:35) = ( − n . If I is a strictly ordered multi-index with values in 1 , . . . , n , put w I = (cid:81) i ∈ I w i , i w I = (cid:81) i ∈ I i w i . Any T ∈ End (cid:0) Λ (cid:0) T ∗ x X (cid:1)(cid:1) can be writtenuniquely as a linear combination of operators of the form w I i w J . Put(12.12) T t = e − iω Xx /t T e iω Xx /t . OHERENT SHEAVES AND RRG 107 Conjugation by the form e − iω Xx /t leaves unchanged the operators w i ∧ and replaces the operators i tw i by i tw i − w i ∧ . Let T max t ∈ C be thecoefficient of ( − i ) n (cid:81) n w i w i in the obvious expansion of T t . By (12.11),(12.12), we get(12.13) Tr sΛ ( T ∗ x X ) [ T ] = t n i n T max t . If T is instead an element of End (cid:0) Λ (cid:0) T ∗ x X (cid:1)(cid:1) (cid:98) ⊗ End ( D s,x ), we still define T t as in (12.12). Equation (12.13) is replaced by(12.14) Tr sΛ ( T ∗ x X ) (cid:98) ⊗ D s,x [ T ] = t n i n Tr s D s,x [ T t ] max . When viewing w i , w i as differential forms on X at x , then ( − i ) n (cid:81) n w i w i is just the canonical volume 2 n -form.In L t,x , no matrix operator of the form w i ∧ appears. Put(12.15) M t,x = e − iω Xx /t L t,x e iω Xx /t . The above conjugation will play the role of Getzler rescaling [G86] inlocal index theory.Let R t,x ( Y, Y (cid:48) ) , Y, Y (cid:48) ∈ T R ,x X be the smooth kernel for exp ( − M t,x )with respect to the volume dY (cid:48) (2 π ) n . By (12.10), (12.14), we get(12.16) Tr s [ P t ( x, x )] = 1 i n Tr s D s,x [ R t,x (0 , max . To compute the asymptotics of (12.16) as t → 0, we use the secondequation in (10.55) for A p ∗ E , t while implementing the above trivializa-tions and conjugations. We will not write the conjugations explicitly.We view R T Xx as a section of Λ (cid:0) T ∗ R ,x X (cid:1) ⊗ End ( T x X ). Let Γ T X bethe connection form for ∇ T X in the above parallel transport trivializa-tion. It is well-known that for Y ∈ T R ,x X ,(12.17) Γ T XY = 12 R T Xx ( Y, · ) + O (cid:0) | Y | (cid:1) . By (12.17), we deduce that if Γ Λ ( T ∗ X ) is the corresponding connectionform for ∇ Λ ( T ∗ X ), then(12.18) Γ Λ ( T ∗ X ) Y = − (cid:68) R T Xx ( Y, · ) w i , w j (cid:69) w i i w j + O (cid:0) | Y | (cid:1) , with O (cid:0) | Y | (cid:1) still being linear combinations of the w i i w j . 08 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI Let Γ D be the connection form in the considered trivialization of D .Then(12.19) − F √ t t (cid:18) ∇ Λ ( T ∗ X ) (cid:98) ⊗ De i − 12 Tr T X [ A ( e i )] + c t i e i C (cid:19) F / √ t = − (cid:32) ∇ e i ( √ tY ) + √ t (cid:18) Γ Λ ( T ∗ X ) + Γ D (cid:19) √ tY (cid:16) e i ( √ tY ) (cid:17) − √ t T X [ A ( e i )] √ tY + √ t c t i e i C √ tY (cid:33) . We conjugate (12.19) by e − iω Xx /t . Using (12.18), we find that as t → ∇ e i ( √ tY ) + √ t (cid:18) Γ Λ ( T ∗ X ) + Γ D (cid:19) √ tY (cid:16) e i ( √ tY ) (cid:17) → ∇ e i + 12 (cid:68) R T Xx ( Y, e i ) w i , w j (cid:69) w i w j . Note that c t B = B , and c t B ∗ is obtained from B ∗ by replacing the ex-terior variables w i by − i tw i . By the considerations that follow (12.11),when conjugating − i tw i by e − iω Xx /t , we obtain w i − i tw i , so that as t → c t i e i C √ tY → i e i C x . By (12.21), we deduce that(12.22) √ t c t i e i C √ tY → . By (12.20), (12.22), we find that after conjugation, as t → − t (cid:18) ∇ Λ ( T ∗ X ) (cid:98) ⊗ De i − 12 Tr T X [ A ( e i )] + c t i e i C (cid:19) → − (cid:18) ∇ e i + 12 (cid:68) R T Xx ( Y, e i ) w i , w j (cid:69) w i w j (cid:19) . We will use the fact that ∂ X ∂ X ω X = 0, and also (10.43). Recall that w i , w j are considered as standard differential forms on X at x . Then(12.23) can be rewritten in the form(12.24) − t (cid:18) ∇ Λ ( T ∗ X ) (cid:98) ⊗ De i − 12 Tr T X [ A ( e i )] + c t i e i C (cid:19) → − (cid:18) ∇ e i + 12 (cid:10) R T Xx Y, e i (cid:11)(cid:19) . OHERENT SHEAVES AND RRG 109 Consider now the remaining terms in the right-hand side of (10.55),while still using the fact that ∂ X ∂ X ω X = 0. By proceeding as before,we find that after conjugation,(12.25) c t (cid:18) A E , + 12 Tr (cid:2) R T X (cid:3)(cid:19) √ tY → (cid:18) A E , + 12 Tr (cid:2) R T X (cid:3)(cid:19) s,x . To obtain the behavior as t → β replaced by β/t , c by c t , and e i by √ te i . After conju-gation, for k ≥ 3, the limit of the corresponding terms vanishes. Since β is a 3-form, i e i i e i β is a 1-form, whose square vanishes. Ultimately,after conjugation, the limit of all the terms vanishes.Put(12.26) M ,s,x = − (cid:18) ∇ e i + 12 (cid:10) R T Xx Y, e i (cid:11)(cid:19) + A E , s,x + 12 Tr (cid:2) R T Xx (cid:3) . By (10.55), (12.24), (12.25), and by the considerations that follow, wesee that as t → M t,x → M ,s,x , in the sense that we have a uniform convergence of the coefficients of theconsidered operators together with all their derivatives over compactsubsets of T R ,x X .Let R ,s,x ( Y, Y (cid:48) ) , Y, Y (cid:48) ∈ T R ,x X denote the smooth kernel associatedwith the operator exp ( − M ,s,x ) with respect to dY (cid:48) (2 π ) n . By proceedingas in [B86, Theorem 4.12], [BGV92, Theorem 10.23] using (12.27), as t → 0, we have the uniform convergence,(12.28) R t,x (0 , → R ,s,x (0 , . Note that Tr s D s,x [ R ,x (0 , T ∗ C S ) (cid:98) ⊗ Λ ( T ∗ C X ),to which the full operator ϕ now applies. Let (cid:8) ϕ Tr D s,x [ Q ,s,x (0 , (cid:9) max ∈ Λ ( T ∗ C S ) be the coefficient of the volume form dx viewed as a 2 n formon T R ,x X . Using (12.16), (12.28), as t → 0, we have the uniformconvergence over X ,(12.29) ϕ (2 π ) − n Tr s [ P t ( x, x )] → (cid:8) ϕ Tr D s,x [ R ,s,x (0 , (cid:9) max . In the right-hand side of (12.27) one recognizes the classical harmonicoscillator in index theory [G86], [BGV92, Proposition 4.19]. UsingMehler’s formula [GlJ87, Theorem 1.5.10] and (8.1), we get(12.30) ϕ Tr D s,x [ R ,s,x (0 , q ∗ (cid:104) (cid:98) A (cid:0) T R X, ∇ T R X (cid:1) e c ( T X,g TX ) / (cid:105) ch (cid:0) A E (cid:48)(cid:48) , g D (cid:1) . 10 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI Using (12.4), (12.29), (12.30), and dominated convergence, we get(12.1). If ω X is closed, ∇ T R X = ∇ T R X , so that (12.1) gives (12.2).If ω X is closed, (12.3) follows from (12.2). We will now prove (12.3)when we only assume that ∂ X ∂ X ω X = 0. Recall that if α ∈ Ω (=) ( X, R )is closed, { α } denotes its Bott-Chern cohomology class in H (=)BC ( X, R ).By Theorem 11.10 and by (12.1), we get(12.31)ch BC ( Rp ∗ E ) = p ∗ (cid:104) q ∗ (cid:110) (cid:98) A (cid:0) T R X, ∇ T R X (cid:1)(cid:111) q ∗ e c , BC ( T X ) ch BC ( E ) (cid:105) . Now we proceed as in [B13, Theorem 5.2.1]. We have the equivalentexact sequence of holomorphic vector bundles on X in (10.41), (10.42).Using the results of Bismut-Gillet-Soul´e [BGS88a, Theorem 1.29] onBott-Chern classes, from the exact sequence in (10.41), using the no-tation in (2.27), we get(12.32) (cid:110) (cid:98) A (cid:0) T R X, ∇ T R X (cid:1)(cid:111) = (cid:98) A BC ( T X ) . By (2.28), we obtain(12.33) Td BC ( T X ) = (cid:98) A BC ( T X ) e c , BC ( T X ) / . By (12.31)–(12.33), we get (12.3). The proof of our theorem is com-pleted. (cid:3) Remark . It is not possible to deform geometrically the exact se-quence (10.42) by scaling ω X . However, on the exact sequence (10.41),the deformation arguments of [BGS88a] can still be used. More pre-cisely, that if d ∈ C ∗ , replacing ω X by dω X in the construction of ∇ F (cid:48)(cid:48) in (10.40) does not change the Bott-Chern class, which ultimately al-lows us to make d = 0, which is equivalent to what we did in the proofof Theorem 12.1. Similar ideas will be used in a more sophisticatecontext in the rest of the paper.13. The hypoelliptic superconnections The purpose of this Section is to construct an antiholomorphic su-perconnection over S with fiberwise hypoelliptic curvature. This super-connection is a deformation of the elliptic superconnections of Section10. Our constructions extend what was done in [B13, Chapter 6] forholomorphic vector bundles on M to the case of antiholomorphic su-perconnections A E (cid:48)(cid:48) .This Section is organized as follows. In Subsection 13.1, we intro-duce the total space X of an extra copy (cid:100) T X of T X , and also theantiholomorphic superconnection N X = π ∗ C X on X . OHERENT SHEAVES AND RRG 111 In Subsection 13.2, given a Hermitian metric g (cid:100) T X on (cid:100) T X , we obtaina corresponding splitting of N X .In Subsection 13.3, we introduce the total space π : M → M of q ∗ (cid:100) T X , and we construct a natural infinite-dimensional antiholomorphicsuperconnection E M = E (cid:98) ⊗ b π ∗ C M over M .In Subsection 13.4, we introduce the superconnection A (cid:48)(cid:48) that corre-sponds to A E M (cid:48)(cid:48) via a natural splitting of E M .In Subsection 13.5, given metrics g (cid:100) T X , g D on (cid:100) T X, D , we constructa nonpositive Hermitian form on the diagonal bundle associated with E M , and the corresponding adjoint superconnection A (cid:48) of A (cid:48)(cid:48) .In Subsection 13.6, if q : M → X is the obvious projection, given aholomorphic section z of q ∗ T X on M , we construct an antiholomorphicsuperconnection A E M (cid:48)(cid:48) z on Λ ( T ∗ M ) (cid:98) ⊗ E M .In Subsection 13.7, still using the above splittings, we obtain a su-perconnection A (cid:48)(cid:48) Z and its companion A (cid:48) Z , which also depends on thechoice of a K¨ahler form ω X on X .In Subsection 13.8, we construct a Hermitian form (cid:15) X along the fibers X .In Subsection 13.9, we compute the proper adjoint of A Z .In Subsection 13.10, we give a Lichnerowicz formula for the curvature A Z .In Subsection 13.11, we specialize our constructions with z is thecanonical section y , in which case A Z is self-adjoint, and its curvatureis fiberwise hypoelliptic.In Subsection 13.12, we study the behavior of A Y under the scalingof g (cid:100) T X .Finally, in Subsection 13.13, we give the arguments which show thatwhen replacing g (cid:100) T X by b g (cid:100) T X , if g T X = g (cid:100) T X , as b → 0, the hypoellipticsuperconnection A Y,b is a deformation of A p ∗ E .In this Section, we make the same assumptions as in Section 10, andwe use the corresponding notation.13.1. The total space of (cid:100) T X . Let X be a compact complex manifold.Let π : X → X be the total space of T X , the fiber being denoted (cid:100) T X to distinguish it from the usual tangent bundle T X , so that T X and (cid:100) T X are canonically isomorphic. The conjugate bundle to (cid:100) T X will bedenoted (cid:100) T X and its dual (cid:91) T ∗ X . The real bundle associated with (cid:100) T X is denoted by (cid:91) T R X . 12 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI We have the exact sequences of holomorphic vector bundles on X ,0 → π ∗ (cid:100) T X → T X → π ∗ T X → , (13.1) 0 → π ∗ T ∗ X → T ∗ X → π ∗ (cid:91) T ∗ X → . Recall that the trivial antiholomorphic superconnection C X was de-fined in (5.12). We define C X on X in the same way. Then(13.2) C X = π ∗ b C X . Put(13.3) N X = π ∗ C X . Then N X is a Λ (cid:0) T ∗ X (cid:1) -module. Let N X be the infinite-dimensionalvector bundle associated with N X . The operator ∂ X can be viewedas an antiholomorphic superconnection A N X (cid:48)(cid:48) on N X , so that N X = (cid:0) N X , A N X (cid:48)(cid:48) (cid:1) .Put(13.4) I = C ∞ (cid:16)(cid:100) T X, π ∗ Λ (cid:16) (cid:91) T ∗ X (cid:17)(cid:17) . Then I is the diagonal vector bundle associated with N X . Also I isthe vector bundle on X of smooth complex functions along the fibre (cid:100) T X , and(13.5) I = Λ (cid:16) (cid:91) T ∗ X (cid:17) ⊗ I . A Hermitian metric on (cid:100) T X . Here, we follow [B08, Section3.12] and [B13, Section 6.1]. The constructions given in these referenceswill permit us to construct a splittings of N X .Let g (cid:100) T X be a Hermitian metric on (cid:100) T X , let ∇ (cid:100) T X be the holomorphicHermitian connection on (cid:16)(cid:100) T X, g (cid:100) T X (cid:17) , and let R (cid:100) T X be its curvature.Let ∇ Λ (cid:18) (cid:91) T ∗ X (cid:19) be the induced connection on Λ (cid:16) (cid:91) T ∗ X (cid:17) .Let T H X ⊂ T X be the horizontal subbundle associated with theconnection ∇ (cid:100) T X . We have the identification of smooth vector bundleson X ,(13.6) T H X = π ∗ T X, and also identification of smooth vector bundles on X ,(13.7) T X = T H X ⊕ π ∗ (cid:100) T X. OHERENT SHEAVES AND RRG 113 By (13.6), (13.7), we get the smooth identification,(13.8) T X = π ∗ (cid:16) T X ⊕ (cid:100) T X (cid:17) . By (13.8), we get the identification of smooth vector bundles,(13.9) Λ (cid:0) T ∗ X (cid:1) = π ∗ (cid:16) Λ (cid:0) T ∗ X (cid:1) (cid:98) ⊗ Λ (cid:16) (cid:91) T ∗ X (cid:17)(cid:17) . By (13.9), we get the identification of smooth vector bundles on X ,(13.10) N X = Λ (cid:0) T ∗ X (cid:1) (cid:98) ⊗ I . By (13.9), we have the identity(13.11) Ω , • ( X , C ) = Ω , • ( X, I • ) . Let I c denote the vector bundle of elements of I which have compactsupport. Then I c inherits a L Hermitian metric g I c from the metric g (cid:100) T X , such that if r, t ∈ I c , then(13.12) (cid:104) r, t (cid:105) g I c = (cid:90) (cid:92) T R X (cid:104) r, t (cid:105) g Λ ( (cid:92) T ∗ X ) dY (2 π ) n . If U ∈ T R X , let U H ∈ T H R X denote the horizontal lift of U .If U ∈ T R X , if s is a smooth section of I on X , set(13.13) ∇ I U s = ∇ Λ (cid:18) (cid:91) T ∗ X (cid:19) U H s. Then ∇ I induces a unitary connection ∇ I c on I c that preserves its Z -grading. Let ∇ I be the restriction of ∇ I to I . By (13.5), the con-nection ∇ I is just the connection on I that is induced by ∇ Λ (cid:18) (cid:91) T ∗ X (cid:19) , ∇ I .Let ∂ V denote the ∂ -operator acting along the fibers (cid:100) T X of π . Using(13.11), we see that ∇ I (cid:48)(cid:48) and ∂ V act on Ω , • ( X , C ).Let (cid:98) y be the tautological section of π ∗ (cid:100) T X on X , let (cid:98) y be the conjugatesection of π ∗ (cid:100) T X , and let (cid:98) Y = (cid:98) y + (cid:98) y be the tautological section of π ∗ (cid:91) T R X , so that (cid:12)(cid:12)(cid:12) (cid:98) Y (cid:12)(cid:12)(cid:12) g (cid:92) T R X = 2 | (cid:98) y | g (cid:100) TX .Let w , . . . , w n be a basis of T X , let w , . . . , w n be the correspondingdual basis of T ∗ X . We denote with a hat the corresponding objectsassociated with (cid:100) T X, (cid:91) T ∗ X . We have results established in [BGS88b,Theorem 2.8], [B13, Propositions 6.1.2, 6.1.4, and 6.1.6]. Proposition 13.1. The following identities hold: ∇ I (cid:48)(cid:48) , = 0 , ∇ I (cid:48) , = 0 , (13.14) ∇ I , = −∇ VR (cid:100) TX (cid:98) Y − (cid:68) R (cid:100) T X (cid:98) w i , (cid:98) w j (cid:69) (cid:98) w i i (cid:98) w j . 14 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI Also (cid:104) ∇ I , ∂ V (cid:105) = 0 , (cid:104) ∇ I , ∂ V ∗ (cid:105) = 0 . (13.15) Finally, we have the identity of operators acting on Ω , • ( X , C ) , (13.16) ∂ X = ∇ I (cid:48)(cid:48) + ∂ V . Remark . The decomposition (13.16) can be viewed as a specialcase of equation (5.6) when E is replaced by N X . In (13.16), ∂ X doesnot depend on g (cid:100) T X , but the splitting in the right-hand depends on g (cid:100) T X .By the above, it follows that (cid:0) I c , ∇ I c (cid:48)(cid:48) (cid:1) is a holomorphic Hermitianvector bundle . The holomorphic structure depends explicitly on themetric g (cid:100) T X .13.3. The antiholomorphic superconnection E M . We make thesame assumptions as in Subsections 10.1, 10.2, and 13.1. Put(13.17) M = S × X . Let π be the projection M → M , let p, q be the obvious maps M → S, M → X .As in (13.2), we have(13.18) C M = π ∗ b C M . Put(13.19) N M = π ∗ C M . Then(13.20) N M = q ∗ b N X . Let N M be the infinite-dimensional vector bundle on M that is associ-ated with N M .The associated diagonal vector bundle is just q ∗ I .Let E = (cid:0) E, A E (cid:48)(cid:48) (cid:1) be an antiholomorphic superconnection on M .Put(13.21) E M = π ∗ b E . Then E M = (cid:16) E M , A E M (cid:48)(cid:48) (cid:17) is an antiholomorphic superconnection on M , and the associated diagonal bundle is just π ∗ D . Also E M is aΛ (cid:0) T ∗ M (cid:1) -module.Put(13.22) E M = π ∗ E M . Here, ‘holomorphic vector bundle’ is taken in the naive sense. OHERENT SHEAVES AND RRG 115 Recall that the tensor product (cid:98) ⊗ b was defined in (5.77). Then(13.23) E M = E (cid:98) ⊗ b N M . Observe that E M = (cid:0) E M , A E M (cid:48)(cid:48) (cid:1) is an antiholomorphic superconnectionon M , with an infinite-dimensional E M , and the associated diagonalbundle is D (cid:98) ⊗ q ∗ I .13.4. A metric description of A E M (cid:48)(cid:48) . We fix a splitting of E as in(5.2), (5.3), so that E (cid:39) E .We proceed as in Subsections 13.2 and 13.3. We fix a metric g (cid:100) T X onthe vector bundle (cid:100) T X over X . This induces a corresponding splittingof N X , and also of N M .The corresponding vector bundles E , N M, are given by E = Λ (cid:0) T ∗ M (cid:1) (cid:98) ⊗ D, N M, = Λ (cid:0) T ∗ M (cid:1) (cid:98) ⊗ q ∗ I . (13.24)The above two splittings induce a corresponding splitting of E M , sothat(13.25) E M, = Λ (cid:0) T ∗ M (cid:1) (cid:98) ⊗ D (cid:98) ⊗ q ∗ I . Equation (13.25) can also be written in the form(13.26) E M, = E (cid:98) ⊗ q ∗ I . Definition 13.3. Let A (cid:48)(cid:48) be the antiholomorphic superconnection on E M, that corresponds to A E M (cid:48)(cid:48) .We write A E (cid:48)(cid:48) as in (5.9). In particular, D is equipped with anantiholomorphic connection ∇ D (cid:48)(cid:48) .As we saw in Subsection 13.2, I is equipped with a holomorphicstructure ∇ I (cid:48)(cid:48) . Definition 13.4. Let ∇ D (cid:98) ⊗ q ∗ I (cid:48)(cid:48) be the antiholomorphic connection on D (cid:98) ⊗ q ∗ I associated with ∇ D (cid:48)(cid:48) , ∇ I (cid:48)(cid:48) . Proposition 13.5. The following identity holds: (13.27) A (cid:48)(cid:48) = ∇ D (cid:98) ⊗ q ∗ I (cid:48)(cid:48) + B + ∂ V . Proof. Our proposition follows from equations (5.9), (13.16). (cid:3) The adjoint of A (cid:48)(cid:48) . Let g D be a Hermitian metric on D . Re-call that g (cid:100) T X induces a Hermitian metric on I c . Therefore D (cid:98) ⊗ q ∗ I c isequipped with a Hermitian metric g D (cid:98) ⊗ q ∗ I c .We are now in a situation formally similar to what was done in afinite-dimensional context in Section 7, and in an infinite-dimensionalcontext in [BGS88b], in [B13, Chapter 4], and in Section 10, with 16 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI M, S replaced by M , M , and p replaced by π . We briefly explain theconstruction of the adjoint of A (cid:48)(cid:48) .Let s, s (cid:48) ∈ Ω (cid:0) M, D (cid:98) ⊗ q ∗ I c (cid:1) . We can write s, s (cid:48) in the form s = (cid:88) α i r i , s (cid:48) = (cid:88) β j t j , (13.28)with α i , β j ∈ Ω ( M, C ) and r i , t j ∈ C ∞ (cid:0) M, D (cid:98) ⊗ q ∗ I c (cid:1) . As in (7.10),(7.12), put(13.29) θ g D (cid:98) ⊗ q ∗ I c ( s, s (cid:48) ) = i m (2 π ) m (cid:88) (cid:90) M (cid:101) α i ∧ β j (cid:104) r i , t j (cid:105) g D (cid:98) ⊗ q ∗ I c . Definition 13.6. Let A (cid:48) be the formal adjoint of A (cid:48)(cid:48) with respect to θ g D (cid:98) ⊗ q ∗ I c .Then A (cid:48) is exactly the adjoint of the antiholomorphic superconnec-tion A (cid:48)(cid:48) in the sense of Definition 7.4 with respect to the pure metric g D (cid:98) ⊗ q ∗ I c .Recall that the holomorphic connection ∇ D (cid:48) on D was defined in(7.15). Since g D is a pure metric in M D , ∇ D (cid:48) is just a standard holo-morphic connection. Definition 13.7. Let ∇ D (cid:98) ⊗ q ∗ I (cid:48) be the holomorphic connection on D (cid:98) ⊗ q ∗ I induced by ∇ D (cid:48) , ∇ I (cid:48) .We use the notation in (7.17). Let ∂ V ∗ denote the formal adjoint of ∂ V with respect to the metric g I c . Proposition 13.8. The following identity holds: (13.30) A (cid:48) = ∇ D (cid:98) ⊗ q ∗ I (cid:48) + B ∗ + ∂ V ∗ . Proof. This is a consequence of Proposition 13.5. (cid:3) Then A (cid:48)(cid:48) , A (cid:48) act as differential operators on Ω (cid:0) M, D (cid:98) ⊗ q ∗ I (cid:1) , and A (cid:48)(cid:48) = 0 , A (cid:48) = 0 . (13.31)13.6. The antiholomorphic superconnection A E M (cid:48)(cid:48) z . Recall that A E M (cid:48)(cid:48) , A E M (cid:48)(cid:48) are the same operator, that acts on C ∞ (cid:0) M , E M (cid:1) = C ∞ ( M, E M ). We extend these operators to operators acting on C ∞ (cid:0) M , π ∗ Λ ( T ∗ M ) (cid:98) ⊗ E M (cid:1) = C ∞ (cid:0) M, Λ ( T ∗ M ) (cid:98) ⊗ E M (cid:1) , that still verify Leibniz rule with respect tomultiplication by smooth forms in π ∗ Λ ( T ∗ M ) , Λ ( T ∗ M ).Let z be a holomorphic section of q ∗ T X on M . Since q ∗ T X ⊂ T M ,the contraction operator i z acts on the above vector spaces. OHERENT SHEAVES AND RRG 117 Definition 13.9. Put A E M (cid:48)(cid:48) z = A E M (cid:48)(cid:48) + i z , A E M (cid:48)(cid:48) z = A E M (cid:48)(cid:48) + i z . (13.32)Both operators have total degree +1. Since z is holomorphic, we get A E M (cid:48)(cid:48) , z = 0 , A E M (cid:48)(cid:48) , z = 0 . (13.33)Also A E M (cid:48)(cid:48) z is an antiholomorphic superconnection on π ∗ Λ ( T ∗ M ) (cid:98) ⊗ E M ,and A E M (cid:48)(cid:48) z is an antiholomorphic superconnection on Λ ( T ∗ M ) (cid:98) ⊗ E M .These two operators are in fact the same operator. In the sequel, wewill deal mostly with the second one.13.7. The superconnection A Z . We now make the same assump-tions as in Subsections 13.4 and 13.6.As in Subsection 10.3, let g T X be a Hermitian metric on T X withK¨ahler form ω X . The exponential e − iω X is taken in the exterior algebraΛ ( T ∗ C X ).Let z denote a holomorphic section of q ∗ T X on M . Put(13.34) Z = z + z. Then Z is a smooth section of q ∗ T R X . Let z ∗ ∈ q ∗ T ∗ X correspond to z ∈ q ∗ T X by the metric g T X . By (10.7), z ∗ can also be viewed as asection of T ∗ M .Note that T X ⊂ T M , and so the contraction operators i z , i z act asoperators of degree 1 , − T ∗ C M ), and so they act on Ω (cid:0) M, D (cid:98) ⊗ q ∗ I (cid:1) .Now we follow [B13, Section 6.3]. Definition 13.10. Put A (cid:48)(cid:48) Z = A (cid:48)(cid:48) + i z , A (cid:48) Z = e q ∗ iω X ( A (cid:48) + i z ) e − q ∗ iω X , A Z = A (cid:48)(cid:48) Z + A (cid:48) Z . (13.35)Since z is holomorphic, A (cid:48)(cid:48) Z = 0 , A (cid:48) Z = 0 . (13.36)By (13.35), we get(13.37) A (cid:48) Z = A (cid:48) − q ∗ ∂ X iω X + i z + z ∗ ∧ . Also(13.38) A Z = [ A (cid:48)(cid:48) Z , A (cid:48) Z ] . Let ∇ D (cid:98) ⊗ q ∗ I be the connection on D (cid:98) ⊗ q ∗ I which is induced by ∇ D , ∇ I .Recall that C was defined in (7.21). By (13.27), (13.30), (13.35), and 18 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI (13.37), we get A (cid:48)(cid:48) Z = ∇ D (cid:98) ⊗ q ∗ I (cid:48)(cid:48) + B + ∂ V + i z , A (cid:48) Z = ∇ D (cid:98) ⊗ q ∗ I (cid:48) + B ∗ − q ∗ ∂ X iω X + ∂ V ∗ + i z + z ∗ ∧ , (13.39) A Z = ∇ D (cid:98) ⊗ q ∗ I + C − q ∗ ∂ X iω X + ∂ V + i z + ∂ V ∗ + i z + z ∗ ∧ . We denote by L the operator of multiplication by ω X and by Λ itsadjoint with respect to g T X . Let w . . . , w n be an orthonormal basis of (cid:0) T X, g T X (cid:1) . Then L = − iw i ∧ w i , Λ = ii w i i w i . (13.40)Conjugation by exp ( i Λ) changes w i , w i into w i − i w i , w i + i w i .Put B (cid:48)(cid:48) Z = exp ( i Λ) A (cid:48)(cid:48) Z exp ( − i Λ) , B (cid:48) Z = exp ( i Λ) A (cid:48) Z exp ( − i Λ) , (13.41) B Z = exp ( i Λ) A Z exp ( − i Λ) . Then(13.42) B Z = B (cid:48)(cid:48) Z + B (cid:48) Z . Set(13.43) E = exp ( i Λ) (cid:16) ∇ D (cid:98) ⊗ q ∗ I + C − q ∗ ∂ X iω X (cid:17) exp ( − i Λ) . By (13.39), and by the considerations that follow (13.40), we get(13.44) B Z = E + ∂ V + i z + ∂ V ∗ + z ∗ . We still count the degree in Λ ( T ∗ C M ) as the difference of the antiholo-morphic and of the holomorphic degree. Then A (cid:48)(cid:48) Z , B (cid:48)(cid:48) Z are operators oftotal degree 1, and A (cid:48) Z , B (cid:48) Z are operators of total degree − M .13.8. Two Hermitian forms. Here we follow [B13, Sections 6.5 and6.6]. We will slightly modify the construction of the Hermitian form θ g D (cid:98) ⊗ q ∗ I .Set(13.45) F = q ∗ (cid:16) Λ ( T ∗ C X ) (cid:98) ⊗ Λ (cid:16) (cid:91) T ∗ X (cid:17)(cid:17) (cid:98) ⊗ D. Then F is a vector bundle on M . Recall that Λ ( T ∗ C X ) is Z -gradedby the difference of the antiholomorphic and the holomorphic degrees. OHERENT SHEAVES AND RRG 119 Also Λ (cid:16) (cid:91) T ∗ X (cid:17) and D are Z -graded. Then F inherits a corresponding Z -grading. Similarly Λ ( T ∗ C S ) inherits a similar grading.As in [B13, Section 6.5], we will use the identification(13.46) Λ (cid:16) (cid:91) T ∗ C X (cid:17) = Λ ( T ∗ X ) (cid:98) ⊗ Λ (cid:16) (cid:91) T ∗ X (cid:17) . By (13.45), (13.46), we get(13.47) F = q ∗ (cid:16) Λ (cid:0) T ∗ X (cid:1) (cid:98) ⊗ Λ (cid:16) (cid:91) T ∗ C X (cid:17)(cid:17) (cid:98) ⊗ D. Let σ be the involution of X given by ( x, (cid:98) y ) → ( x, − (cid:98) y ). Then σ induces an involution of M .Let σ ∗ denote the obvious action of σ on Λ (cid:16) (cid:91) T ∗ C X (cid:17) . Namely, if N Λ ( (cid:92) T ∗ C X ) is the number operator of Λ (cid:16) (cid:91) T ∗ C X (cid:17) , then σ ∗ acts like ( − N Λ ( (cid:92) T ∗ C X ).Also, we make σ ∗ act trivially on D . Then σ ∗ acts on F in (13.47).We fix one fiber X of the projection p , and the corresponding fiber X of p . Note that(13.48) Ω (cid:0) X, D | X (cid:98) ⊗ I (cid:1) = C ∞ ( X , π ∗ F | X ) . Let dv X denote the volume form on X which is associated with themetrics g T X , g (cid:100) T X . If s, s (cid:48) ∈ C ∞ , c ( X , π ∗ F | X ), set(13.49) (cid:104) s, s (cid:48) (cid:105) L = (2 π ) − n (cid:90) X (cid:104) s, s (cid:48) (cid:105) g F dv X . Definition 13.11. If s, s (cid:48) ∈ C ∞ , c ( X , π ∗ F | X ), put(13.50) η X ( s, s (cid:48) ) = (cid:104) σ ∗ s, s (cid:48) (cid:105) L . Then η X is a Hermitian form on C ∞ , c ( X , π ∗ F | X ). Note that η X isnon-degenerate, it is positive on σ -invariant forms, and negative on σ -anti-invariant forms. Also the C ∞ , c ( X , π ∗ F i | X ) are mutually orthog-onal with respect to (cid:104) (cid:105) L and η X .Let σ ∗ be the restriction of σ ∗ to Λ (cid:16) (cid:91) T ∗ X (cid:17) . We make σ ∗ act triviallyon Λ ( T ∗ C X ) and on D . By (13.45), σ ∗ also acts on F . The differencewith σ ∗ is that σ ∗ acts trivially on Λ ( T ∗ X ). Then σ ∗ acts on D | X (cid:98) ⊗ I .Let s, s (cid:48) ∈ Ω (cid:0) X, D | X (cid:98) ⊗ I c (cid:1) . Then s, s (cid:48) can be written in the form s = (cid:88) α i r i , s (cid:48) = (cid:88) β j t j , (13.51)with α i , β j ∈ Ω ( X, C ), and r i , t j ∈ C ∞ (cid:0) X, D | X (cid:98) ⊗ I c (cid:1) .If α ∈ Λ ( T ∗ C X ), we define (cid:101) α as in (4.33). Also we use the samenotation as in Subsection 8.2. As we observed in that Subsection, 20 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI multiplication by iω X is a self-adjoint operator with respect to theform θ in (7.1). Also the exponential e − iω X is taken in Λ ( T ∗ C X ). Definition 13.12. Put(13.52) (cid:15) X ( s, s (cid:48) ) = i n (2 π ) n (cid:90) X (cid:101) α i ∧ e − iω X β j (cid:104) σ ∗ r, t (cid:105) g D | X (cid:98) ⊗ I c . Equation (13.52) can be rewritten in the form,(13.53) (cid:15) X ( s, s (cid:48) ) = i n (2 π ) n (cid:90) X (cid:101) α i ∧ e − iω X β j (cid:104) σ ∗ r, t (cid:105) g D | X (cid:98) ⊗ Λ ( (cid:92) T ∗ X ) dY (2 π ) n . By [B13, Section 6.5 and Theorem 6.6.1], (cid:15) X is a Hermitian formon C ∞ , c ( X , π ∗ F | X ) = Ω (cid:0) X, I c (cid:98) ⊗ D | X (cid:1) . As explained in [B13, Sec-tion 6.6], if instead, we replace integration over X by integration over M , while replacing i n (2 π ) n by i m (2 π ) m , we obtain a Hermitian form (cid:15) M onΩ (cid:0) M, D (cid:98) ⊗ q ∗ I c (cid:1) . Remark . As explained in Subsection 8.2, the term e − iω X in (13.52)can be interpreted as a generalized metric on the trivial line C over M .Equivalently, e − iω X g D | X can be viewed as a generalized metric on D | X .By [B13, eq. (6.5.10) and Theorem 6.6.1], if s, s (cid:48) ∈ Ω (cid:0) X, D | X (cid:98) ⊗ q ∗ I c (cid:1) = C ∞ ( X , π ∗ F | X ) , we get(13.54) (cid:15) X ( s, s (cid:48) ) = η X (cid:0) e i Λ s, e i Λ s (cid:48) (cid:1) . Then (cid:15) X is also a non-degenerate Hermitian form, which can also beproved directly. By (13.54), or by direct inspection, the C ∞ , c ( X , π ∗ F i | X )are mutually orthogonal with respect to (cid:15) X .We will denote with an upper † the adjoint of an operator withrespect to (cid:15) X or η X , the corresponding Hermitian form being explicitlymentioned.We have the following result established in [B13, Proposition 6.5.1and Theorem 6.6.1]. Proposition 13.14. If adjoints are taken with respect to (cid:15) X , if e ∈ T X, f ∈ T ∗ X , then i † e = − e ∗ ∧ − i e , i † e = e ∗ ∧ − i e , (13.55) ( f ∧ ) † = − f ∧ , (cid:0) f ∧ (cid:1) † = − f ∧ . If e ∈ T X, E = e + e ∈ T R X , then e ∗ ∧ − i E , e ∗ ∧ + i E are skew-adjointwith respect to (cid:15) X . OHERENT SHEAVES AND RRG 121 Self-adjointness of A Z , B Z . Note that p ∗ (cid:0) Λ ( T ∗ M ) (cid:98) ⊗ E M (cid:1) is aΛ ( T ∗ C S )-module.We will use the conventions of Subsection 10.3 when taking adjointsof antiholomorphic superconnections over the base S . These adjointswill be taken with respect to the Hermitian forms (cid:15) X or η X , and willbe denoted with a † . The adjoints of A Z and of its components A (cid:48)(cid:48) Z , A (cid:48) Z are taken with respect to (cid:15) X , and the adjoints of B Z and of its com-ponents B (cid:48)(cid:48) Z , B (cid:48) Z are taken with respect to η X . In the case of A Z or ofits components, the adjoint could be taken as well with respect to thestandard Hermitian form (cid:15) M .Let z − be the section of q ∗ T X given by(13.56) z − ( x, Y ) = − z ( x, − Y ) . Then z − is still holomorphic. Let Z − be the corresponding section of q ∗ T R X . Theorem 13.15. The operators A (cid:48)(cid:48) Z , B (cid:48)(cid:48) Z are of total degree , and theoperators A (cid:48) Z , B (cid:48) Z are of total degree − . Moreover, we have the iden-tities A (cid:48)(cid:48)† Z = A (cid:48) Z − , B (cid:48)(cid:48)† Z = B (cid:48) Z − , (13.57) A † Z = A Z − , B † Z = B Z − . Proof. The proof is the same as in [B13, Theorem 6.5.2]. For A Z andits components, it is an easy consequence of (13.52) and of Proposition13.14. (cid:3) Remark . When considering A Z and its components, we may takeadjoints with respect to (cid:15) M , and the above results would still hold. Ad-joints with respect to (cid:15) M are simply classical adjoints. Using integrationalong the fiber p ∗ to evaluate (cid:15) M is enough to explain the coincidenceof the adjoints.13.10. A formula for the curvature of A Z . When identifying T X and (cid:100) T X , we denote by (cid:98) g T X the metric on T X corresponding to g (cid:100) T X ,and by (cid:98) ∇ T X the connection on T X that corresponds to ∇ (cid:100) T X . Equiv-alently, (cid:98) ∇ T X is the Chern connection on (cid:0) T X, (cid:98) g T X (cid:1) . Let (cid:98) τ be thetorsion of (cid:98) ∇ T X . Then (cid:98) τ has the same properties as τ . Recall that Z = z + z is a smooth section on M of q ∗ T R X . Let ρ : M → X be the obvious projection. We identify Z with its horizontallift Z H in ρ ∗ T H R X ⊂ T R M . The torsion (cid:98) τ takes its values in T X and not in (cid:100) T X . 22 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI We denote by (cid:98) ∇ T R X,H Z the section on M of π ∗ ( T ∗ R M ⊗ q ∗ T R X ) thatis obtained by taking the covariant derivative of Z with respect to thepull-back of (cid:98) ∇ T R X along horizontal directions in M . Other horizontalcovariant derivatives will be denoted in a similar way.Let ∆ Vg (cid:100) TX be the Laplacian along the fibers (cid:100) T X with respect to themetric g (cid:100) T X .In this Subsection, † refers to the adjoint with respect to the form (cid:15) X in Definition 13.12. By Proposition 13.14 and by (13.56), we get(13.58) i † z − = z ∗ ∧ + i z . Now we follow Subsection 4.6. For a ∈ R , we introduce the connec-tion (cid:98) ∇ T X,a on T X associated with (cid:98) ∇ T X . If (cid:98) ∇ Λ ( T ∗ R X ) ,a is the inducedconnection on Λ ( T ∗ R X ), by (4.49), if U ∈ T R X , then(13.59) (cid:98) ∇ Λ ( T ∗ R X ) ,aU = (cid:98) ∇ Λ ( T ∗ R X ) U − ai (cid:98) τ ( U, · ) . The connection (cid:98) ∇ Λ ( T ∗ R X ) ,a preserves the complex type of the forms. Inparticular, by (4.52), we get(13.60) d X = (cid:98) ∇ Λ ( T ∗ R X ) , − / , and (13.60) splits into formulas for ∂ X , ∂ X . Equation (13.60) can beextended to an equation for d M ,(13.61) d M = (cid:98) ∇ Λ ( T ∗ R X ) , − / . In the right-hand side of (13.61), exterior differentiation on S is doneimplicitly, and does not require the choice of a connection on T S .Recall that F was defined in (13.45). Let (cid:98) ∇ F be the connection on F induced by (cid:98) ∇ Λ ( T ∗ C X ) , ∇ D , ∇ Λ (cid:18) (cid:91) T ∗ X (cid:19) .For a ∈ R , let (cid:98) ∇ F ,a be the connection on F induced by (cid:98) ∇ Λ ( T ∗ C X ) ,a , ∇ D , ∇ Λ (cid:18) (cid:91) T ∗ X (cid:19) .By (13.59), we get(13.62) (cid:98) ∇ F ,aU = (cid:98) ∇ F U − ai (cid:98) τ ( U, · ) . By (4.51), we obtain(13.63) (cid:98) ∇ F , − / = (cid:98) ∇ F a + i (cid:98) τ . In [B13], the connection ∇ T X on T X was used instead. This accounts forminor differences with respect to [B13]. OHERENT SHEAVES AND RRG 123 As explained after (13.61), equation (13.63) can be viewed as an iden-tity of operators acting on Ω (cid:16) M, q ∗ Λ (cid:16) (cid:91) T ∗ X (cid:17) (cid:98) ⊗ D (cid:17) . Equation (13.63)can be rewritten in the form(13.64) (cid:98) ∇ F , − / = ∇ q ∗ Λ (cid:18) (cid:91) T ∗ X (cid:19)(cid:98) ⊗ D , where the right-hand side is simply the natural extension of a con-nection on the vector bundle q ∗ Λ (cid:16) (cid:91) T ∗ X (cid:17) (cid:98) ⊗ D to an operator acting onΩ (cid:16) M, q ∗ Λ (cid:16) (cid:91) T ∗ X (cid:17) (cid:98) ⊗ D (cid:17) . Definition 13.17. Let (cid:98) ∇ F be the fibrewise connection on F such thatif U ∈ T R X , U = u + u, u ∈ T X ,(13.65) (cid:98) ∇ F U = (cid:98) ∇ F , − U − i u q ∗ ∂ X iω X . Put(13.66) F = q ∗ Λ ( T ∗ C X ) (cid:98) ⊗ D (cid:98) ⊗ q ∗ I . By (13.5), (13.45), we get(13.67) F = F (cid:98) ⊗ q ∗ I . Equation (13.67) can be rewritten in the form,(13.68) F = C ∞ (cid:16) (cid:91) T R X, π ∗ F | (cid:92) T R X (cid:17) . Let (cid:98) ∇ F be the connection on F associated with (cid:98) ∇ Λ ( T ∗ C X ) , ∇ D , ∇ I .Equivalently, (cid:98) ∇ F is the connection on F induced by (cid:98) ∇ F , ∇ I . Moregenerally, when replacing F by F , we obtain corresponding objectsassociated with F .As in Subsection 4.6, (cid:98) ∇ F ,a a denotes the antisymmetrization of (cid:98) ∇ F ,a .By (13.39), (13.64) can be rewritten in the form A (cid:48)(cid:48) Z = (cid:98) ∇ F , − / (cid:48)(cid:48) a + B + ∂ V + i z , A (cid:48) Z = (cid:98) ∇ F , − / (cid:48) a + B ∗ − q ∗ ∂ X iω X + ∂ V ∗ + i z + z ∗ ∧ , (13.69) A Z = (cid:98) ∇ F , − / + C − q ∗ ∂ X iω X + ∂ V + i z + ∂ V ∗ + i z + z ∗ ∧ . Recall that the curvature A E , was defined in Subsection 7.2. It isa section of degree 0 of Λ ( T ∗ C M ) (cid:98) ⊗ End ( D ).Let w , . . . , w n be a basis of T X . The corresponding basis of (cid:100) T X is denoted (cid:98) w , . . . , (cid:98) w n . In the sequel, we assume that (cid:98) w , . . . , (cid:98) w n is anorthonormal basis of (cid:16)(cid:100) T X, g (cid:100) T X (cid:17) . As before, if e ∈ T R X , e ∗ ∈ T ∗ R X corresponds to e via the metric g T R X . 24 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI We establish an extension of [B13, Theorem 6.8.1]. Theorem 13.18. The following identity holds: (13.70) A Z = (cid:104) ∂ V + i z , ∂ V † + i † z − (cid:105) + (cid:104) (cid:98) ∇ F , − / (cid:48)(cid:48) a + B, i † z − (cid:105) + (cid:104) (cid:98) ∇ F , − / (cid:48) a + B ∗ − q ∗ ∂ X iω X , i z (cid:105) − q ∗ ∂ X ∂ X iω X − ∇ VR (cid:100) TX (cid:98) Y − (cid:68) R (cid:100) T X (cid:98) w i , (cid:98) w j (cid:69) (cid:98) w i i (cid:98) w j + A E , . Equation (13.70) can be rewritten in the form, (13.71) A Z = (cid:104) ∂ V + i z , ∂ V † + i † z − (cid:105) + (cid:98) ∇ F Z + i Z C − q ∗ ∂ X ∂ X iω X − ∇ VR (cid:100) TX (cid:98) Y − (cid:68) R (cid:100) T X (cid:98) w i , (cid:98) w j (cid:69) (cid:98) w i i (cid:98) w j + A E , + i (cid:98) ∇ T R X,H Z + ∇ T X (cid:48)(cid:48) ,H z ∗ . Moreover, (13.72) (cid:104) ∂ V + i z , ∂ V † + i † z − (cid:105) = 12 (cid:16) − ∆ Vg (cid:100) TX + | Z | g TX (cid:17) + (cid:98) w i ∧ (cid:16) i ∇ (cid:98) wi z + ∇ (cid:98) w i z ∗ (cid:17) − i (cid:98) w i i ∇ (cid:98) wi z . Proof. Since (cid:98) σ ∗ commutes with ∂ V , we get(13.73) ∂ V ∗ = ∂ V † . Equation (13.70) follows from Proposition 13.1 and from equations(13.38), (13.58), (13.69), and (13.73). Equation (13.71) is an easy con-sequence. The only mysterious point is the that in (cid:98) ∇ F , the connection (cid:98) ∇ F , − appears, while there was (cid:98) ∇ F , − / in equation (13.70). But thisis entirely explained in (4.56)–(4.58).The proof of (13.72) is left to the reader, which completes the proofof our theorem. (cid:3) The hypoelliptic curvature. Among the sections z , we willdistinguish the tautological section y of q ∗ T X on M corresponding tothe tautological section (cid:98) y of (cid:100) T X . Note that y, (cid:98) y already appeared inSubsection 13.2. The section y is such that(13.74) y − = y. We use the notation(13.75) Y = y + y. OHERENT SHEAVES AND RRG 125 By (13.69), we get A (cid:48)(cid:48) Y = (cid:98) ∇ F , − / (cid:48)(cid:48) a + B + ∂ V + i y , A (cid:48) Y = (cid:98) ∇ F , − / (cid:48) a + B ∗ − q ∗ ∂ X iω X + ∂ V ∗ + i y + y ∗ ∧ , (13.76) A Y = (cid:98) ∇ F , − / + C − q ∗ ∂ X iω X + ∂ V + i y + ∂ V ∗ + i y + y ∗ ∧ . By Theorem 13.15, we get A (cid:48)(cid:48)† Y = A (cid:48) Y , A † Y = A Y . (13.77)Put(13.78) γ = ∇ T X − (cid:98) ∇ T X . Then γ is a section of T ∗ X ⊗ End ( T X ). If u ∈ T X , then γu ∗ is a(1 , X . Definition 13.19. Let (cid:98) ∇ F be the fiberwise connection on F , which issuch that if u ∈ T X, U = u + u ∈ T R X , then(13.79) (cid:98) ∇ F U = (cid:98) ∇ F U + γu ∗ . As before, we may as well replace F by F and obtain a fiberwiseconnection (cid:98) ∇ F on F . Theorem 13.20. The following identity holds: (13.80) A Y = (cid:104) ∂ V + i y , ∂ V † + i † y (cid:105) + (cid:104) (cid:98) ∇ F , − / (cid:48)(cid:48) a + B, i † y (cid:105) + (cid:104) (cid:98) ∇ F , − / (cid:48) a + B ∗ − q ∗ ∂ X iω X , i y (cid:105) − q ∗ ∂ X ∂ X iω X − ∇ VR (cid:100) TX (cid:98) Y − (cid:68) R (cid:100) T X (cid:98) w i , (cid:98) w j (cid:69) (cid:98) w i i (cid:98) w j + A E , . Equation (13.80) can be rewritten in the form (13.81) A Y = (cid:104) ∂ V + i y , ∂ V † + i † y (cid:105) + (cid:98) ∇ F Y + i Y C − q ∗ ∂ X ∂ X iω X − ∇ VR (cid:100) TX (cid:98) Y − (cid:68) R (cid:100) T X (cid:98) w i , (cid:98) w j (cid:69) (cid:98) w i i (cid:98) w j + A E , . Moreover, (13.82) (cid:104) ∂ V + i y , ∂ V † + i † y (cid:105) = 12 (cid:16) − ∆ Vg (cid:100) TX + | Y | g TX (cid:17) + (cid:98) w i ∧ ( i w i + w i ∗ ) − i (cid:98) w i i w i . Proof. This is a consequence of Theorem 13.18. In the proof of (13.81),we have exploited the fact that (cid:98) ∇ T R X,H Y = 0, and the related equation(13.83) ∇ T X (cid:48)(cid:48) ,H y ∗ = γy ∗ . 26 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI (cid:3) By Theorem 13.18 and by H¨ormander [H67], A Y is a hypoellipticoperator along the fibers X . More precisely, A Y is a hypoelliptic Lapla-cian in the sense of [B05, B08, B12]. Remark . In (13.81), we may as well replace (cid:98) ∇ F Y by (cid:98) ∇ F Y H .13.12. Scaling the metric g (cid:100) T X . Here we follow [B13, Section 7.1].For b > 0, when g (cid:100) T X is replaced by b g (cid:100) T X , let (cid:15) X,b , (cid:15) M,b , A (cid:48) Y,b , A Y,b bethe analogues of (cid:15) X , (cid:15) M , A (cid:48) Y , A Y . Definition 13.22. For a > 0, let δ a be the dilation of M : ( x, Y ) → ( x, aY ). For a > 0, let δ ∗ a be the action of δ a on smooth sections ofΛ (cid:16) (cid:91) T ∗ X (cid:17) . The action of δ ∗ a extends to F .Also y/b is a holomorphic section on M of q ∗ T X . One verifies easilythat(13.84) A Y/b = δ ∗ /b A Y,b δ ∗− /b . This identity is obvious because δ ∗ /b maps the metric b g (cid:100) T X to g (cid:100) T X ,and Y to Y /b .Now we follow [B13, Section 6.9]. For a > 0, put(13.85) K a s ( x, Y ) = s ( x, aY ) . Definition 13.23. Put C Y,b = K b A Y/b K − b , D Y,b = K b B Y/b K − b . (13.86)By (13.41), (13.86), we get(13.87) D Y,b = exp ( i Λ) C Y,b exp ( − i Λ) . By (13.39), (13.44), we get C Y,b = ∇ D (cid:98) ⊗ q ∗ I + C − q ∗ ∂ X iω X + 1 b (cid:16) ∂ V + i y + ∂ V ∗ + i y + y ∗ (cid:17) , (13.88) D Y,b = E + 1 b (cid:16) ∂ V + i y + ∂ V ∗ + y ∗ (cid:17) . Hypoelliptic and elliptic superconnections. In this Sub-section, we assume that (cid:98) g T X = g T X . Let w , . . . , w n be an orthonormalbasis of T X with respect to g T X , let (cid:98) w , . . . , (cid:98) w n be the correspondingbasis of (cid:100) T X . We denote the dual basis in the usual way. OHERENT SHEAVES AND RRG 127 We use the identification in (13.46). Set (cid:98) L = − iw j ∧ (cid:98) w j , (cid:98) Λ = ii (cid:98) w j i w j . (13.89)In (13.89), (cid:98) L is just the multiplication operator by the fibrewise K¨ahlerform (cid:98) ω X ,V . As in [B13, eq. (6.9.8)], we get(13.90) (cid:104) ∂ V + i y , ∂ V ∗ + y ∗ ∧ (cid:105) = 12 (cid:16) − ∆ Vg TX + | Y | g TX (cid:17) − i (cid:16)(cid:98) L − (cid:98) Λ (cid:17) . The right-hand side of (13.90) is a harmonic oscillator.Let S (0 , • ) (cid:16)(cid:100) T X, π ∗ Λ ( T ∗ X ) (cid:17) be the Schwartz space of rapidly de-creasing sections of π ∗ (cid:16) Λ (cid:16) (cid:91) T ∗ X (cid:17) (cid:98) ⊗ Λ ( T ∗ X ) (cid:17) along the fibre (cid:100) T X . Theoperator in (13.90) is essentially self-adjoint on S (0 , • ) (cid:16)(cid:100) T X, π ∗ Λ ( T ∗ X ) (cid:17) .By [B90, Proposition 1.5 and Theorem 1.6] , [B13, Section 6.9], itsspectrum is N , and its kernel is 1-dimensional and spanned by(13.91) δ = exp (cid:16) i (cid:98) ω X ,V − | Y | g TX / (cid:17) . Recall that dY is the volume form along the fibers (cid:100) T X with respect to g (cid:100) T X . Note that σ ∗ δ = δ, (2 π ) − n (cid:90) (cid:100) T X | δ | dY = 1 . (13.92)Let P be the L orthogonal projection operator on vector space spannedby δ .We embed D = Ω , • ( X, D | X ) into Ω (cid:0) X, D | X (cid:98) ⊗ I (cid:1) by the embedding α → π ∗ α ∧ δ , so that D is just the L kernel of ∂ V + i y + ∂ V ∗ + y ∗ ∧ .More generally, we embed p ∗ E into p ∗ (cid:2) E (cid:98) ⊗ q ∗ (cid:0) Λ ( T ∗ X ) (cid:98) ⊗ I (cid:1)(cid:3) by thesame formula.Recall that the elliptic superconnection A p ∗ E was defined in Defini-tion 10.8, and that the differential operator E was defined in (13.43).Now we give an extension of [B13, Theorem 6.9.2]. Theorem 13.24. The following identity holds: (13.93) P E P = A p ∗ E . Proof. Observe that (13.93) is an identity of differential operators over M , which we will prove by local methods on M . We will use theconsiderations that follow (13.40) to evaluate E . In [B90], the operator ∂ V + i √− y and its adjoint were considered instead,which explains the minor discrepancies with respect to what is done here. 28 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI Let ∇ Λ( T ∗ X ) be the connection induced by ∇ T X on Λ ( T ∗ X ). Since g T X = (cid:98) g T X , we get(13.94) ∇ Λ( T ∗ X ) (cid:98) ⊗ I δ = 0 . Recall that C = B + B ∗ . As explained after equation (13.40), conju-gation by exp ( i Λ) changes w i into w i + i w i . Recall that δ in (13.91) isof total degree 0. The effect of the composition of the i w j is to map δ into forms of total positive degree , which are orthogonal to δ . Using(13.94) and proceeding as in the proof of [B13, Theorem 6.9.2], we findthat(13.95) P exp ( i Λ) (cid:16) ∇ D (cid:98) ⊗ q ∗ I (cid:48)(cid:48) + B (cid:17) exp ( − i Λ) P = ∇ D (cid:48)(cid:48) + B. In [B13], the proof of (13.95) uses equations (13.63), (13.76).Using (10.15), we can rewrite (13.95) in the form(13.96) P exp ( i Λ) (cid:16) ∇ D (cid:98) ⊗ q ∗ I (cid:48)(cid:48) + B (cid:17) exp ( − i Λ) P = A p ∗ E (cid:48)(cid:48) . Conjugation by exp ( i Λ) changes w i into w i − i w i . By proceedingagain as in [B13, Theorem 6.9.2], we get(13.97) P exp ( i Λ) (cid:16) ∇ D (cid:98) ⊗ q ∗ I (cid:48) + B ∗ − q ∗ ∂ X iω X (cid:17) exp ( − i Λ) P = ∇ D (cid:48) ,S + ∇ D (cid:48)(cid:48) ,X ∗ α + c B ∗ . By (10.22), we can rewrite (13.97) in the form,(13.98) P exp ( i Λ) (cid:16) ∇ D (cid:98) ⊗ q ∗ I (cid:48) + B ∗ − q ∗ ∂ X iω X (cid:17) exp ( − i Λ) P = A p ∗ E (cid:48) . By (13.96), (13.98), we get (13.93). The proof of our theorem iscompleted. (cid:3) Remark . Recall that D Y,b was defined in (13.87), and that it splitsas(13.99) D Y,b = D (cid:48)(cid:48) Y,b + D (cid:48) Y,b . By (13.57), when taking adjoints with respect to η X , then(13.100) D (cid:48)(cid:48)† Y,b = D (cid:48) Y,b . By (13.50), (13.92), on Ω , • ( X, D | X ), η X restrict to the Hermitianproduct (cid:104) (cid:105) L . Since the contribution of ∇ D (cid:98) ⊗ q ∗ I (cid:48)(cid:48) + B to P E P is A p ∗ E (cid:48)(cid:48) ,the contribution of its adjoint ∇ D (cid:98) ⊗ q ∗ I (cid:48) + B ∗ − q ∗ ∂ X iω X has to be A p ∗ E (cid:48) .This explains some aspects of the proof of Theorem 13.24. Recall that the total degree is the difference of the antiholomorphic and of theholomorphic degree. OHERENT SHEAVES AND RRG 129 The hypoelliptic superconnection forms The purpose of this Section is to construct the superconnection formsassociated with the hypoelliptic superconnections of Section 13, and toprove that their class in Bott-Chern cohomology coincides with theclass of the elliptic superconnection forms.This Section is organized as follows. In Subsection 14.1, we con-struct the hypoelliptic superconnection forms, that depend on metrics g T X , g (cid:100) T X , g D .In Subsection 14.2, when g T X = (cid:98) g T X , we show that when replacing g (cid:100) T X by b g (cid:100) T X , as b → 0, the hypoelliptic superconnection forms con-verge to the elliptic superconnection forms, which gives the main resultof this Section.In this Section, we make the same assumptions as in Sections 10 and13, and we use the corresponding notation.14.1. The hypoelliptic forms. We will say that two metrics g (cid:100) T X , g (cid:100) T X (cid:48) lie in the same projective class if there exists a constant c > g (cid:100) T X (cid:48) = cg (cid:100) T X . Let p be such a projective class. Once we fix g (cid:100) T X ∈ p ,then p (cid:39) R ∗ + .We fix p . Let Q p be the collection of parameters ω X , g D , g (cid:100) T X , with g (cid:100) T X ∈ p . We denote by d Q p the de Rham operator on Q p . A key factis that when g (cid:100) T X ∈ p , then ∇ I does not depend on g (cid:100) T X .By [BL08, Section 3.3], the operator exp ( −A Y ) is trace class, and isgiven by a smooth kernel along the fibers X that is rapidly decreasingalong the fibre (cid:100) T X together with its derivatives, and this uniformlyover S . Definition 14.1. Put(14.1) ch (cid:16) A (cid:48)(cid:48) Y , ω X , g D , g (cid:100) T X (cid:17) = ϕ Tr s (cid:2) exp (cid:0) −A Y (cid:1)(cid:3) . The Hermitian form (cid:15) X was defined in Definition 13.12, and dependson the splitting of E , and also on ω X , g D , g (cid:100) T X .Given p , (cid:15) − X d Q p (cid:15) X is a 1-form on Q p with values in (cid:15) X -self-adjointendomorphisms. It can be easily computed from equation (13.52).We establish an extension of [B13, Theorem 7.3.2]. Theorem 14.2. The form ch (cid:16) A (cid:48)(cid:48) Y , ω X , g D , g (cid:100) T X (cid:17) lies in Ω (=) ( S, R ) ,it is closed, and its Bott-Chern cohomology class does not depend on ω X , g D , g (cid:100) T X , or on the splitting of E . 30 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI Given p , then (14.2) d Q p ch (cid:16) A (cid:48)(cid:48) Y , ω X , g D , g (cid:100) T X (cid:17) = − ∂ S ∂ S iπ Tr s (cid:2) (cid:15) − X d Q p (cid:15) X exp (cid:0) −A Y (cid:1)(cid:3) . Proof. The proof that the above forms are closed is formally the sameas the proof of Theorem 10.15. Since A Y is self-adjoint with respect to (cid:15) X , the same argument as in the proof of Theorem 8.2 shows that theabove forms are real. The fact that (cid:15) X is non-positive is irrelevant. Byproceeding as in the proof of Theorem 10.15, i.e., by using deformationover P , one can prove that their corresponding class in Bott-Cherncohomology does not depend on the parameters. Finally, if one restricts g (cid:100) T X to vary in a class p , the proof of (14.2) is the same as the proof ofTheorems 8.2 and 10.15. (cid:3) Definition 14.3. Let ch BC ( A (cid:48)(cid:48) Y ) ∈ H (=)BC ( S, R ) denote the commonclasses in Bott-Chern cohomology of the forms ch (cid:16) A (cid:48)(cid:48) Y , ω X , g D , g (cid:100) T X (cid:17) .14.2. The limit as b → of the hypoelliptic superconnectionforms. In this Subsection, we fix ω X , g D , and we take g (cid:100) T X = b g T X .Then A (cid:48)(cid:48) Y does not depend on b > (cid:0) A p ∗ E (cid:48)(cid:48) , ω X , g D (cid:1) were defined in Definition 10.14, and their Bott-Chern class ch BC (cid:0) A p ∗ E (cid:48)(cid:48) (cid:1) in Definition 10.16. Theorem 14.4. As b → , (14.3) ch (cid:0) A (cid:48)(cid:48) Y , ω X , g D , b g T X (cid:1) → ch (cid:0) A p ∗ E (cid:48)(cid:48) , ω X , g D (cid:1) . Proof. We use the notation of Subsection 13.12. By definition,(14.4) ch (cid:0) A (cid:48)(cid:48) Y , ω X , g D , b g T X (cid:1) = ϕ Tr s (cid:2) exp (cid:0) −A Y,b (cid:1)(cid:3) . By (13.84)–(13.87), we can rewrite (14.4) in the form(14.5) ch (cid:0) A (cid:48)(cid:48) Y , ω X , g D , b g T X (cid:1) = ϕ Tr s (cid:2) exp (cid:0) −D Y,b (cid:1)(cid:3) . Using (13.88), Theorem 13.24, and by proceeding exactly as in theproofs of [BL08, Theorem 5.2.1], [B08, Theorem 7.13], and [B13, The-orem 7.6.2], we get (14.3). The proof of our theorem is completed. (cid:3) Theorem 14.5. The following identities hold: (14.6) ch BC ( A (cid:48)(cid:48) Y ) = ch BC (cid:0) A p ∗ E (cid:48)(cid:48) (cid:1) = ch BC ( Rp ∗ E ) in H (=)BC ( S, R ) . Proof. This is an obvious consequence of Theorems 11.10, 14.2, and14.4. (cid:3) OHERENT SHEAVES AND RRG 131 The hypoelliptic superconnection forms when ∂ X ∂ X ω X = 0The purpose of this Section is to prove Theorem 10.1 when ∂ X ∂ X ω X =0 using the hypoelliptic superconnection forms of Section 14. This re-sult had already been proved in Section 12 using elliptic superconnec-tion forms. While the hypoelliptic superconnection forms do not givea better result, the techniques used in the present Section prepare forSection 16, where Theorem 10.1 will be established in full generalityusing deformations of our hypoelliptic superconnections.This Section is organized as follows. In Subsection 15.1, we computefinite and infinite-dimensional supertraces on certain model operators.These computations will later be used in a local index theoretic context.In Subsection 15.2, given t > 0, when replacing (cid:16) ω X , g (cid:100) T X , g D (cid:17) by (cid:16) ω X /t, g (cid:100) T X /t , g D (cid:17) , we compute the corresponding hypoelliptic curva-ture, and we establish various scaling identities.Finally, in Subsection 15.3, when ∂ X ∂ X ω X = 0, we obtain the limitas t → Finite and infinite-dimensional traces. Let W be a complexvector space of dimension n , and let W R be the corresponding realvector space. If W C = W R ⊗ R C is its complexification, then W C = W ⊕ W .Let W be another copy of W . Observe that(15.1) Λ (cid:16) W ⊕ W ∗ (cid:17) = Λ (cid:0) W (cid:1) (cid:98) ⊗ Λ (cid:16) W ∗ (cid:17) . Let w , . . . , w n be a basis of W , let w , . . . , w n be the associateddual basis of W ∗ . Let w , . . . , w n be the corresponding basis of W ,and let w , . . . , w n the associated dual basis of W ∗ . The supertracemaps Λ (cid:16) W ⊕ W ∗ (cid:17) (cid:98) ⊗ End (cid:16) Λ (cid:16) W ∗ (cid:17)(cid:17) into Λ (cid:16) W ⊕ W ∗ (cid:17) and vanisheson supercommutators.Note that Λ n (cid:16) W ∗ (cid:17) (cid:98) ⊗ Λ n (cid:0) W (cid:1) is canonically trivial, and α = (cid:81) n w i w i is the canonical section. If θ ∈ Λ (cid:16) W ∗ (cid:17) (cid:98) ⊗ Λ (cid:0) W (cid:1) , let θ max ∈ C be thecoefficient of α in the expansion of θ .If A ∈ End ( W ), then we still denote by A the conjugate action of A on W .We establish a form of a result of Mathai-Quillen [MQu86, eq. (2.13)],[B08, Proposition 6.10]. 32 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI Proposition 15.1. If A ∈ End ( W ) , then (15.2)Tr s (cid:2) exp (cid:0) − w i w i + i w j w j + (cid:10) Aw i , w j (cid:11) w i i w j (cid:1)(cid:3) max = Td − (cid:0) − A (cid:1) . Proof. If h ∈ W , let [ h ] denote the element that corresponds in W . Asimilar notation is used for elements in W ∗ .First, we assume that A is invertible. Let (cid:101) A ∈ End ( W ∗ ) be thetranspose of A . Put(15.3) M = − w i i A − w i − w i (cid:104) (cid:101) A − w i (cid:105) . If f ∈ W, u ∈ W ∗ , thenexp ( M ) u exp ( − M ) = u − (cid:104) (cid:101) A − u (cid:105) , (15.4) exp ( M ) i f exp ( − M ) = i f − (cid:2) A − f (cid:3) . By (15.4), we deduce that(15.5) exp ( M ) (cid:10) Aw i , w j (cid:11) w i i w j exp ( − M ) = (cid:10) Aw i , w j (cid:11) w i i w j − w i w i + i w j w j + (cid:10) A − w i , w j (cid:11) w i w j . Since supertraces vanish on supercommutators, we deduce from (15.5)that(15.6) Tr s (cid:2) exp (cid:0) − w i w i + i w j w j + (cid:10) Aw i , w j (cid:11) w i i w j (cid:1)(cid:3) = exp (cid:0) − (cid:10) A − w i , w j (cid:11) w i w j (cid:1) Tr s (cid:2) exp (cid:0)(cid:10) Aw i , w j (cid:11) w i i w j (cid:1)(cid:3) . Also,(15.7) Tr s (cid:2) exp (cid:0)(cid:10) Aw i , w j (cid:11) w i i w j (cid:1)(cid:3) = det (cid:0) − exp (cid:0) A (cid:1)(cid:1) . If A is diagonalizable, we get(15.8) (cid:2) exp (cid:0) − (cid:10) A − w i , w j (cid:11) w i w j (cid:1)(cid:3) max = (cid:2) det (cid:0) − A (cid:1)(cid:3) − . Equation (15.8) extends by continuity when A is only supposed tobe invertible. By (15.6)–(15.8), we get (15.2) when A is invertible.This equation extends by continuity to arbitrary A . The proof of ourproposition is completed. (cid:3) Let (cid:99) W be another copy of W . Let g (cid:99) W be a Hermitian metric on (cid:99) W , and let (cid:98) g W be the corresponding metric on W . Let ( U, Y ) be thegeneric element of W R ⊕ (cid:99) W R . Let dU dY be the associated volume formon W R ⊕ (cid:99) W R . Let ∆ (cid:99) W R be the Laplacian on (cid:99) W R . Let F ∈ End (cid:16)(cid:99) W (cid:17) be skew-adjoint. Then F acts as antisymmetric endomorphism of (cid:99) W R . OHERENT SHEAVES AND RRG 133 Let ∇ (cid:99) W R F Y denote differentiation along the vector field F Y . Let ∇ W R Y denote differentiation along W R in the direction Y .Let p be the scalar operator,(15.9) p = − 12 ∆ (cid:99) W R + ∇ (cid:99) W R F Y + ∇ W R Y . Let s (( U, Y ) , ( U (cid:48) , Y (cid:48) )) denote the smooth kernel associated with exp ( − p )with respect to the volume dU (cid:48) dY (cid:48) (2 π ) n .Now we state a result taken from [B08, Theorem 6.8]. Theorem 15.2. If | F | < π , the function s ((0 , Y ) , (0 , Y )) is inte-grable, and we have the identity, (15.10) (cid:90) (cid:99) W R s ((0 , Y ) , (0 , Y )) dY (2 π ) n = (cid:98) A ( F | W R ) = Td ( F | W ) Td ( − F | W ) . Proof. We will give a self-contained proof of (15.10). Let p ,ξ be theFourier transform of p in the variable U . If ξ ∈ W ∗ R (cid:39) W R , then(15.11) p ,ξ = − 12 ∆ (cid:99) W R + ∇ (cid:99) W R F Y + 2 iπ (cid:104) ξ, Y (cid:105) . Assume that F is invertible. Put(15.12) q ,ξ = e iπ (cid:104) ξ,F − Y (cid:105) p ,ξ e − iπ (cid:104) ξ,F − Y (cid:105) . Since F is skew-adjoint, we get(15.13) q ,ξ = − 12 ∆ (cid:99) W R + ∇ (cid:99) W R F Y − iπF − ξ + 2 π (cid:12)(cid:12) F − ξ (cid:12)(cid:12) Let r ,ξ be the operator deduced from q ,ξ by the translation Y → Y + 2 iπF − ξ . Then(15.14) r ,ξ = − 12 ∆ (cid:99) W R + ∇ (cid:99) W R F Y + 2 π (cid:12)(cid:12) F − ξ (cid:12)(cid:12) . Since F is skew-adjoint, the operators in the right-hand side of (15.14)are commuting, and the smooth kernel r ,ξ ( Y, Y (cid:48) ) for exp ( − r ,ξ ) withrespect to dY (cid:48) (2 π ) n is given by(15.15) r ,ξ ( Y, Y (cid:48) ) = exp (cid:18) − π (cid:12)(cid:12) F − ξ (cid:12)(cid:12) − (cid:12)(cid:12) e − F Y − Y (cid:48) (cid:12)(cid:12) (cid:19) . 34 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI By (15.15), if q ,ξ ( Y, Y (cid:48) ) is the smooth kernel for exp ( − q ,ξ ) with re-spect to dY (cid:48) (2 π ) n , then(15.16) q ,ξ ( Y, Y (cid:48) ) = exp (cid:16) − π (cid:12)(cid:12) F − ξ (cid:12)(cid:12) (cid:17) exp (cid:18) − (cid:12)(cid:12) e − F (cid:0) Y − iπF − ξ (cid:1) − (cid:0) Y (cid:48) − iπF − ξ (cid:1)(cid:12)(cid:12) (cid:19) . By (15.16), we deduce that if 1 − e − F is invertible, then(15.17) (cid:90) (cid:99) W R q ,ξ ( Y, Y ) dY (2 π ) n = exp (cid:16) − π | F − ξ | (cid:17) det (1 − e − F ) | W R . Using (15.17), we deduce that if 1 − e − F is invertible, then(15.18) (cid:90) (cid:99) W R s ((0 , Y ) , (0 , Y )) dY (2 π ) n = det (cid:18) F − e − F (cid:19) | W R , which is equivalent to (15.10). The proof of our theorem is completed. (cid:3) Remark . Using (15.16) and inverting the Fourier transform, onecan give an exact formula for the kernel s (( U, Y ) , ( U (cid:48) , Y (cid:48) )).15.2. The time parameter. Let A Y be the superconnection(15.19) A Y = (cid:98) ∇ F , − / − q ∗ ∂ X iω X + ∂ V + i y + ∂ V ∗ + i y + y ∗ ∧ . By (13.76), we get(15.20) A Y = A Y + C. First, we extend the constructions of [B13, Section 7.1]. Definition 15.4. Let A Y,t be the superconnection A Y associated withthe metrics (cid:16) ω X /t, g D , g (cid:100) T X /t (cid:17) .In the sequel, N, N V , N H , N (cid:98) V denote the total number operators ofΛ ( T ∗ C M ) , Λ ( T ∗ C X ) , Λ ( T ∗ C S ) , Λ (cid:16) (cid:91) T ∗ X (cid:17) . Recall that for a > K a wasdefined in (13.85). We will establish an extension of [B13, Proposition7.1.5]. Here, the number operators are the classical sums of antiholomorphic andholomorphic degrees. OHERENT SHEAVES AND RRG 135 Proposition 15.5. For t > , the following identity holds: (15.21) t N (cid:98) V / N V / K t A Y,t K − t t − N (cid:98) V / − N V / = t − N H / (cid:16) √ t A Y + t N/ Ct − N/ (cid:17) t N H / . Proof. We split A Y,t as in (15.20), i.e.,(15.22) A Y,t = A Y,t + C. The same arguments as in [B13, Proposition 7.1.5], or an easy explicitcomputation using (13.76) show that(15.23) t N (cid:98) V / N V / K t A Y,t K − t t − N (cid:98) V / − N V / = t − N H / √ t A Y t N H / . By (15.20), (15.23), since N H + N V = N , we get (15.21). The proof ofour proposition is completed. (cid:3) Let A tY be the superconnection A Y for the metrics (cid:16) ω T X /t, g D , g (cid:100) T X (cid:17) .The same argument as in the proof of (13.84) shows that(15.24) A tt / Y = δ ∗ t / A Y,t δ ∗− t / . By (15.21), (15.24), we obtain(15.25) K / √ t t N V / A tt / Y t − N V / K √ t = t − N H / (cid:16) √ t A Y + t N/ Ct − N/ (cid:17) t N H / . Set(15.26) M t = A t, t / Y . When ω X is replaced by ω X /t , the connections (cid:98) ∇ F , (cid:98) ∇ F defined in(13.65), (13.79) depend on t , and will be denoted (cid:98) ∇ F t , (cid:98) ∇ F t . A similarnotation will be used for the connections on F .By Theorems 13.18 and 13.20, we get(15.27) M t = − 12 ∆ Vg (cid:100) TX + t | Y | g TX + (cid:98) w i ∧ (cid:0) t / i w i + t / w i ∗ (cid:1) − t / i (cid:98) w i i w i + t / (cid:98) ∇ F t,Y + t / i Y C − q ∗ ∂ X ∂ X iω X /t − ∇ VR (cid:100) TX (cid:98) Y − (cid:68) R (cid:100) T X (cid:98) w i , (cid:98) w j (cid:69) (cid:98) w i i (cid:98) w j + A E , . Observe that δ ∗− t A tt / Y δ ∗ t is the superconnection A √ tY associatedwith the metrics (cid:16) ω X /t, g D , g (cid:100) T X /t (cid:17) .Set(15.28) M (cid:48) t = δ ∗− t M t δ ∗ t . 36 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI By (15.27), we get(15.29) M (cid:48) t = − t Vg (cid:100) TX + 12 | Y | g TX + (cid:98) w i ∧ (cid:0) t / i w i + t − / w i ∗ (cid:1) − t / i (cid:98) w i i w i + t / (cid:98) ∇ F t,Y + t / i Y C − q ∗ ∂ X ∂ X iω X /t − ∇ VR (cid:100) TX (cid:98) Y − (cid:68) R (cid:100) T X (cid:98) w i , (cid:98) w j (cid:69) (cid:98) w i i (cid:98) w j + A E , . Recall that N V is the total number operator of Λ ( T ∗ C X ). Put M (cid:48)(cid:48) t = t N V / M (cid:48) t t − N V / , A E , t = t N V / A E , t − N V / , (15.30) (cid:98) ∇ F t = t N V / (cid:98) ∇ F t t − N V / , C t = t N V / Ct − N V / . Again, in (15.30), we may as well obtain a connection (cid:98) ∇ F t on F .Using (13.65), (13.79), and (15.30), we get(15.31) (cid:98) ∇ F t,U = (cid:98) ∇ F , − U − i u ∂ X iω X + γu ∗ . Since (cid:98) ∇ F t does not depend on t , we will write instead (cid:98) ∇ F , and alsoobtain a corresponding connection (cid:98) ∇ F on F .By (15.29), we obtain(15.32) M (cid:48)(cid:48) t = − t Vg (cid:100) TX + 12 | Y | g TX + (cid:98) w i ∧ ( i w i + w i ∗ ) − t i (cid:98) w i i w i + t / (cid:98) ∇ F Y − i Y C t − tq ∗ ∂ X ∂ X iω X − t ∇ VR (cid:100) TX (cid:98) Y − t (cid:68) R (cid:100) T X (cid:98) w i , (cid:98) w j (cid:69) (cid:98) w i i (cid:98) w j + A E , t . Observe that in (15.32), there are no longer diverging terms as t → (cid:99) dx be the volume form on X with respect to the metric g (cid:100) T X .Given s ∈ S, t > 0, let P t ( z, z (cid:48) ) , P (cid:48)(cid:48) t ( z, z (cid:48) ) be the smooth kernel forexp ( − M t ) , exp ( − M (cid:48)(cid:48) t ) with respect to the volume dv X = (cid:99) dxdY / (2 π ) n on X s .Let d ( x, x (cid:48) ) denote the Riemannian distance on X with respect to (cid:98) g T X .Here is a result inspired by [BL08, Proposition 4.7.1]. Proposition 15.6. There exist m ∈ N , c > , C > such that for s ∈ S, t ∈ ]0 , , if z = ( x, Y ) , z (cid:48) = ( x (cid:48) , Y (cid:48) ) , then (15.33) | P (cid:48)(cid:48) t ( z, z (cid:48) ) | ≤ Ct m exp (cid:16) − c (cid:16) | Y | + | Y (cid:48) | + d ( x, x (cid:48) ) /t (cid:17)(cid:17) . Again, we do not write explicitly the dependence of these kernels on s . OHERENT SHEAVES AND RRG 137 Proof. Comparing equation (15.32) for M (cid:48)(cid:48) t with [BL08, eq. (4.7.4)],and using the fact the above matrix terms are uniformly bounded, theproof is the same as the proof of [BL08, Proposition 4.7.1]. (cid:3) Observe that(15.34) ch (cid:16) A (cid:48)(cid:48) Y , ω X /t, g D , g (cid:100) T X /t (cid:17) = ϕ (cid:90) X s Tr s [ P t ( z, z )] dv X ( z ) . In (15.34), P t can be replaced by P (cid:48)(cid:48) t .By proceeding as in [BL08, Proposition 4.8.2] and in [B08, Propo-sition 6.3], one essential consequence of Proposition 15.6 is that toevaluate the asymptotics of (15.34) as t → 0, we may proceed locally near any x ∈ X , and this uniformly on M = X × S . This means thatgiven x ∈ X , the asymptotics of (cid:82) (cid:92) T R ,x X Tr s [ P t (( x, Y ) , ( x, Y ))] dY canbe evaluated locally near x , and ultimately that X can be suitably re-placed by T R ,x X . This shows that the asymptotics of (15.34) as t → X .15.3. The case when ∂ X ∂ X ω X = 0 .Theorem 15.7. If ∂ X ∂ X ω X = 0 , then (15.35)ch (cid:16) A (cid:48)(cid:48) Y , ω X /t, g D , g (cid:100) T X /t (cid:17) → p ∗ (cid:2) q ∗ Td (cid:0) T X, (cid:98) g T X (cid:1) ch (cid:0) A E (cid:48)(cid:48) , g D (cid:1)(cid:3) . If ∂ X ∂ X ω X = 0 , then (15.36) ch BC ( Rp ∗ E ) = p ∗ [ q ∗ Td BC ( T X ) ch BC ( E )] in H (=)BC ( S, R ) . Proof. We proceed as in [B08, Theorem 6.1] and [B13, Theorem 9.1.1].We start from equation (15.34) which we write in the form,(15.37) ch (cid:16) A (cid:48)(cid:48) Y , ω X /t, g D , g (cid:100) T X /t (cid:17) = ϕ (cid:90) X (cid:34)(cid:90) (cid:92) T R ,x X Tr s [ P t (( x, Y ) , ( x, Y ))] dY (2 π ) n (cid:35) (cid:99) dx (2 π ) n . Put(15.38) m t ( x ) = ϕ π ) n (cid:90) (cid:92) T R ,x X Tr s [ P t (( x, Y ) , ( x, Y ))] dY (2 π ) n , so that(15.39) ch (cid:16) A (cid:48)(cid:48) Y , ω X /t, g D , g (cid:100) T X /t (cid:17) = (cid:90) X m t ( x ) (cid:99) dx. For (cid:15) > x ∈ X , using geodesic coordinateswith respect to the metric (cid:98) g T X centered at x , we can identify the open 38 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI ball B T R X (0 , (cid:15) ) with the corresponding open ball B X ( x, (cid:15) ) in X . Alsoalong the geodesics based at x , we trivialize the vector bundle (cid:100) T X byparallel transport with respect to ∇ (cid:100) T X . We identify the total space of X over B X ( x, (cid:15) ) with B T R ,x X (0 , (cid:15) ) × (cid:92) T R ,x X . Near x , we trivialize F byparallel transport along geodesics with respect to the connection (cid:98) ∇ F , − .This connection is associated with the connections (cid:98) ∇ T X, − , ∇ (cid:100) T X , ∇ D on T X, (cid:100) T X, D . Our operator M t acts now on sections of F x over B T R ,x X (0 , (cid:15) ) × (cid:91) T R X .In principle, we should extend the restriction of the operator M t to B T R ,x X (0 , (cid:15)/ × (cid:92) T R ,x X to the full H x = T R ,x X × (cid:92) T R ,x X . Since ourcoordinate system on X near x is not holomorphic, we cannot simplyview H x as a complex manifold. Still because of the spinor interpre-tation of Λ (cid:0) T ∗ X (cid:1) , we can construct this extension using the meth-ods of [B08, Section 6.6]. Ultimately, we get an operator N t,x actingon C ∞ (cid:0) H x , Λ (cid:0) T ∗ C ,s S (cid:1) (cid:98) ⊗ F x (cid:1) which is still hypoelliptic, and which coin-cides with M t over B T R ,x X (0 , (cid:15)/ × (cid:92) T R ,x X . Let Q t,x (( U, Y ) , ( U (cid:48) , Y (cid:48) ))denote its smooth kernel on H x with respect to the volume dU (cid:48) dY (cid:48) (2 π ) n Byproceeding as in [BL08, Proposition 4.8.2] and in [B08, Theorem 6.1],from the structure of M (cid:48)(cid:48) t , when studying the asymptotics as t → m t ( x ), we may as well replace M t by N t,x . This means that given x ∈ X , we may as well replace m t ( x ) by(15.40) n t ( x ) = ϕ π ) n (cid:90) (cid:92) T R ,x X Tr s [ Q t,x ((0 , Y ) , (0 , Y ))] dY (2 π ) n . For a > 0, let I a be the map acting on smooth functions on H x = T R ,x X × (cid:92) T R ,x X with values in p ∗ Λ (cid:0) T ∗ C ,s S (cid:1) (cid:98) ⊗ F s,x that is given by(15.41) I a s ( U, Y ) = s ( aU, Y ) . Set(15.42) O t,x = I t / N t,x I t − / . Let R t,x (( U, Y ) , ( U (cid:48) , Y (cid:48) )) be the smooth kernel on H x associatedwith the operator exp ( − O t,x ) with respect to the volume dU (cid:48) dY (cid:48) (2 π ) n . By(15.42), we deduce that(15.43) Q t,x ((0 , Y ) , (0 , Y )) = t − n R t,x ((0 , Y ) , (0 , Y )) . By (15.40), (15.43), we obtain(15.44) n t ( x ) = ϕ π ) n t n (cid:90) (cid:92) T R ,x X Tr s [ R t,x ((0 , Y ) , (0 , Y ))] dY (2 π ) n . OHERENT SHEAVES AND RRG 139 We will use the same notation as in Subsection 15.1. Let W beanother copy of T X . Recall that w , . . . , w n is an orthonormal basis of T x X with respect to g (cid:100) T X . Let w , . . . , w n be the corresponding basisof W x . Then w i , w i , ≤ i ≤ n generate the algebra Λ ( W C ,x ).Note that(15.45) Tr sΛ ( T ∗ C ,x X ) (cid:34) n (cid:89) w i i w i w i i w i (cid:35) = 1 , and that the supertrace of the monomials in the above operators hav-ing length less than 2 n vanishes. If T ∈ End (cid:0) Λ (cid:0) T ∗ C ,x X (cid:1)(cid:1) , T can bewritten as a linear combination of normally ordered products of the w i , i w j , w k , i w (cid:96) . If T is a normally ordered monomial, its action on1 ∈ Λ (cid:0) T ∗ C ,x (cid:1) is trivial unless it does not contain operators i w i , i w j , inwhich case it is multiplication by a form in Λ (cid:0) T ∗ C ,x X (cid:1) .Set(15.46) o = n (cid:88) i =1 (cid:0) w i w i + w i w i (cid:1) . Then o does not depend on the choice of the base w . . . , w n .Let T t ∈ Λ ( W C ,x ) (cid:98) ⊗ End (cid:0) Λ (cid:0) T ∗ C ,x X (cid:1)(cid:1) be given by(15.47) T t = exp (cid:0) − o /t / (cid:1) T exp (cid:0) o /t / (cid:1) . Then T t is obtained from T by replacing the operators i w i , i w i by i w i + w i /t / , i w i + w i /t / , while leaving unchanged the w i , w i . Note that T t ∈ Λ ( W C ,x ) (cid:98) ⊗ Λ (cid:0) T ∗ C ,x X (cid:1) .Note that β = (cid:81) n w i w i is a canonical section of the trivial linebundle λ = Λ n ( T ∗ X ) (cid:98) ⊗ Λ n ( W ), and that β = (cid:81) n w i w i is the corre-sponding conjugate section of λ .Let [ T t max ∈ C be the coefficient of ββ in the expansion of T t 1. By(15.45), using the previous considerations, we conclude that(15.48) Tr sΛ ( T ∗ C ,x X ) [ T ] = t n [ T t max . If instead, T is a section of End (cid:0) Λ (cid:0) T ∗ C ,x X (cid:1)(cid:1) (cid:98) ⊗ End (cid:16) Λ (cid:16) (cid:92) T ∗ X x (cid:17) (cid:98) ⊗ D s,x (cid:17) ,we define T t exactly as before. Equation (15.48) takes the form(15.49) Tr sΛ ( T ∗ C ,x X ) (cid:98) ⊗ Λ (cid:18) (cid:91) T ∗ x X (cid:19)(cid:98) ⊗ D s,x [ T ] = t n Tr sΛ (cid:18) (cid:91) T ∗ x X (cid:19)(cid:98) ⊗ D s,x [[ T t max ] . This means that the products of operators i w i , i w j appear to the right of theproducts of operators w i , w j . 40 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI Put(15.50) P t,x = exp (cid:0) − o /t / (cid:1) O t,x exp (cid:0) o /t / (cid:1) . Let S t,x (( U, Y ) , ( U (cid:48) , Y (cid:48) )) be the smooth kernel for the operator exp ( − P t,x )with respect to the volume dU (cid:48) dY (cid:48) (2 π ) n . By (15.44), (15.49), and (15.50), weget(15.51) n t ( x ) = ϕ π ) n (cid:90) (cid:92) T R ,x X Tr sΛ (cid:18) (cid:91) T ∗ x X (cid:19)(cid:98) ⊗ D s,x [[ S t,x ((0 , Y ) , (0 , Y )) 1] max ] dY (2 π ) n . Let ∆ Vg (cid:100) TX be the Laplacian along the fibre (cid:92) T R ,x X with respect to themetric g (cid:100) T X . Let ∇ T R ,x XY be the obvious differentiation operator alongthe fibre T R ,x X .Put(15.52) P ,s,x = − 12 ∆ Vg (cid:100) TX + (cid:98) w i w i − i (cid:98) w i w i − ∇ VR (cid:100) TX Y − (cid:68) R (cid:100) T Xx (cid:98) w i , (cid:98) w j (cid:69) (cid:98) w i i (cid:98) w j + ∇ T R ,x XY + A E , , the tensors are evaluated at ( s, x ).We claim that as in [B08, Theorem 6.7] and in [B13, Theorem 9.1.1],as t → P t,x → P ,s,x , in the sense that the coefficients of the operator P t,x converge uniformlytogether with all their derivatives uniformly over compact sets towardsthe corresponding coefficients of P ,x . To establish (15.53), we will useequation (15.27) for M t , and we will exploit the fact that ∂ X ∂ X ω X = 0.The first two terms in the right-hand side of (15.27) do not causeany problem. Using the considerations that follow (15.47), the limitas t → (cid:98) w i w i − i (cid:98) w i w i .By (13.65), (13.79), we get(15.54) t / (cid:98) ∇ F t,Y = t / (cid:98) ∇ F , − Y + t / (cid:0) − i y ∂ X iω X + γy ∗ (cid:1) . In our given trivialization, at ( U, Y ) ∈ H x , the section of T R ,x X thatcorresponds to the section Y of T R X is just the tautological section Y ∈ T R ,x X . Let Γ (cid:100) T X denote the connection form for (cid:98) ∇ T X in the giventrivialization. In the above coordinate system the vector field Y H , asa section of H x , splits as(15.55) Y H ( U, Y ) = Y ⊕ (cid:16) − Γ (cid:100) T XU ( Y ) Y (cid:17) . OHERENT SHEAVES AND RRG 141 Recall that T X, (cid:100) T X have been trivialized by parallel transport withrespect to (cid:98) ∇ T X, − , ∇ (cid:100) T X . Let (cid:98) Γ T X, − denote the connection form for (cid:98) ∇ T X, − in this trivialization of T X . The lift (cid:98) Γ Λ ( T ∗ R X ) , − of (cid:98) Γ T X, − toΛ ( T ∗ R X ) is given by(15.56) (cid:98) Γ Λ ( T ∗ R X ) , − = − (cid:68)(cid:98) Γ T X, − w i , w j (cid:69) w i i w j − (cid:28)(cid:98) Γ T X, − w i , w j (cid:29) w i i w j . Also, near U = 0,(15.57) (cid:98) Γ T X, − U = O ( U ) . Let d T R ,x X be the de Rham operator on T R ,x X . Let Γ Λ (cid:18) (cid:91) T ∗ X (cid:19) , Γ D be the connection forms associated with ∇ Λ (cid:18) (cid:91) T ∗ X (cid:19) , ∇ D in the giventrivializations. By (15.56), we get(15.58) (cid:98) ∇ F , − = d T R ,x X − (cid:68)(cid:98) Γ T X, − w i , w j (cid:69) w i i w j − (cid:28)(cid:98) Γ T X, − w i , w j (cid:29) w i i w j + Γ Λ (cid:18) (cid:91) T ∗ X (cid:19) + Γ D . By (15.55), (15.58), we get(15.59) (cid:98) ∇ F , − Y = (cid:98) ∇ F , − Y H = ∇ T R ,x XY − ∇ (cid:92) T R ,x X Γ (cid:100) TXU ( Y ) Y − (cid:68)(cid:98) Γ T X, − U ( Y ) w i , w j (cid:69) w i i w j − (cid:28)(cid:98) Γ T X, − U ( Y ) w i , w j (cid:29) w i i w j + Γ Λ (cid:18) (cid:91) T ∗ X (cid:19) U ( Y ) + Γ DU ( Y ) . By (15.58), we get(15.60) I t / t / (cid:98) ∇ F , − Y I t − / = ∇ T R ,x XY − t / ∇ (cid:92) T R ,x X Γ (cid:100) TXt / U ( Y ) Y − (cid:68)(cid:98) Γ T X, − t / U ( Y ) w i , w j (cid:69) t / w i i w j − (cid:28)(cid:98) Γ T X, − t / U ( Y ) w i , w j (cid:29) t / w i i w j + t / Γ Λ (cid:18) (cid:91) T ∗ X (cid:19) t / U ( Y ) + t / Γ Dt / U ( Y ) . 42 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI Let us now consider the effect on (15.60) of the conjugation byexp (cid:0) − o /t / (cid:1) , that only affects the third and fourth terms in the right-hand side. As mentioned before, the terms t / w i i w j , t / w i i w j are re-placed by w i (cid:0) t / i w j + w j (cid:1) , w i (cid:0) t / i w j + w j (cid:1) . Because of (15.57), in(15.60), these terms do not survive as t → t → t / (cid:98) ∇ F , − Y → ∇ T R ,x XY . Let us now consider the second term in the right-hand side of (15.54).This term is a section of Λ ( T ∗ C X ). In the parallel transport trivializa-tion with respect to (cid:98) ∇ T X, − , it remains so. Also it is unaffected by theconjugation by exp (cid:0) − o /t / (cid:1) . Combining (15.54) and (15.61), we findthat as t → 0, after conjugations and rescalings,(15.62) t / (cid:98) ∇ F t,Y → ∇ T R ,x XY . For the same reasons as before, as t → 0, in (15.27), the term t / i Y C disappears. Also we made the assumption that ∂ X ∂ X ω X = 0. The lastthree terms in the right-hand side of (15.27) can be handled exactly asbefore.Putting together the previous considerations gives a proof of (15.53).Let S ,s,x (( U, Y ) , ( U (cid:48) , Y (cid:48) )) be the smooth kernel associated with exp ( − P ,s,x )with respect to the volume dU (cid:48) dY (cid:48) (2 π ) n . By proceeding as in [BL08, Theorem4.10.1], there exist C > , c > t ∈ ]0 , , x ∈ X ,(15.63) | S t,x ((0 , Y ) , (0 , Y )) | ≤ C exp (cid:16) − c | Y | / (cid:17) . By proceeding as in the same reference, and using (15.53), we find thatas t → S t,x (( U, Y ) , ( U (cid:48) , Y (cid:48) )) → S ,s,x (( U, Y ) , ( U (cid:48) , Y )) . By (15.51), (15.63), and (15.64), we find that as t → n t ( x ) → n ,s ( x )= ϕ π ) n (cid:90) (cid:92) T R ,x X Tr sΛ (cid:18) (cid:91) T ∗ x X (cid:19)(cid:98) ⊗ D s,x [[ S ,s,x ((0 , Y ) , (0 , Y )) 1] max ] dY (2 π ) n . By (15.63), there is C > x ∈ X, t ∈ ]0 , | n t ( x ) | ≤ C. OHERENT SHEAVES AND RRG 143 By (15.39), (15.65), and (15.66), we conclude that as t → (cid:16) A (cid:48)(cid:48) Y , ω X /t, g (cid:100) T X /t (cid:17) → (cid:90) X n ,s ( x ) (cid:99) dx. We will evaluate the integrand in the right-hand side of (15.65).Observe that(15.68) ββ = ( − n n (cid:89) w i w i n (cid:89) w i w i . From now on, we will view the w i , w i as sections of Λ (cid:0) T ∗ C ,x X (cid:1) . Then(15.69) n (cid:89) w i w i = ( − i ) n (cid:99) dx. If (cid:15) ∈ Λ ( T ∗ C M ), let (cid:15) (2 n ) ∈ Λ ( T ∗ C M ) be the component of (cid:15) which isof top vertical degree 2 n . Let (cid:15) max ∈ Λ ( T ∗ C S ) be such that(15.70) (cid:15) (2 n ) = (cid:15) max (cid:99) dx. We can identity w i and w i by the metric (cid:98) g T x X . Since R (cid:100) T X takes itsvalues in skew-adjoint endomorphisms of (cid:100) T X , using Proposition 15.1and Theorem 15.2, by (15.65), we get(15.71) n ,s ( x ) = ϕ iπ ) n (cid:104) (cid:98) A (cid:16) R (cid:100) T X (cid:17) Td − (cid:16) R (cid:100) T X (cid:17) Tr s (cid:2) exp (cid:0) − A E , (cid:1)(cid:3)(cid:105) max s,x . From the last identity in (15.10), we get(15.72) (cid:98) A (cid:16) R (cid:100) T X (cid:17) Td − (cid:16) R (cid:100) T X (cid:17) = Td (cid:16) − R (cid:100) T X (cid:17) . By (8.1), (15.71), and (15.72), we obtain(15.73) (cid:90) X n ( x ) (cid:99) dx = p ∗ (cid:2) q ∗ Td (cid:0) T X, (cid:98) g T X (cid:1) ch (cid:0) A E (cid:48)(cid:48) , g D (cid:1)(cid:3) . By combining (15.67) and (15.73), we get (15.35). Using Theorem 14.5and (15.35), we get (15.36). The proof of our theorem is completed. (cid:3) Exotic superconnections andRiemann-Roch-Grothendieck The purpose of this Section is to establish Theorem 10.1, which isthe final step in the proof of our main result. More precisely, we de-fine deformed hypoelliptic superconnections, which when introducing ascaling parameter t > 0, have the proper asymptotics when t → 0, andthis without any assumption on the K¨ahler form. Most of the technical 44 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI tools of local index theory which are needed were already developed inSection 15.This Section is organized as follows. In Subsection 16.1, following[B13, Chapter 11], we introduce a deformation ω Xθ of the K¨ahler form ω X , that depends on θ = ( c, d ), with c ∈ [0 , , d > 0, and coincideswith ω X for c = 0. This deformation ω Xθ also depends on the coordinate Y ∈ (cid:91) T R X . Given a holomorphic section z of q ∗ T X , we construct asuperconnection A θ,Z .In Subsection 16.2, we give a Lichnerowicz formula for A θ,Z .In Subsection 16.3, we take z = y , in which case the curvature is stillhypoelliptic. We define corresponding deformed hypoelliptic supercon-nection forms, and we show that their Bott-Chern class is the same asthe class of our previous hypoelliptic superconnection forms.In Subsection 16.4, we introduce the parameter t > 0, and we estab-lish various scaling identities.In Subsection 16.5, we make c = 1, and we state uniform estimateson the heat kernels associated with the curvature.In Subsection 16.6, we recall a result of [B13] on the explicit con-struction of hypoelliptic superconnection forms of vector bundles.In Subsection 16.7, when c = 1 and with d now depending explic-itly on t , we compute the asymptotics of the deformed hypoellipticsuperconnection forms.Finally, in Subsection 16.8, we prove Theorem 10.1.We make the same assumptions as in Sections 14 and 15, and we usethe corresponding notation.16.1. A deformation of the K¨ahler form ω X . Here, we follow [B13,Chapter 11]. Definition 16.1. If θ = ( c, d ) , ≤ c ≤ , d > 0, put(16.1) ω Xθ = (cid:18) − c + cd | Y | g (cid:100) TX (cid:19) ω X . Equivalently,(16.2) ω Xθ = ω X + c (cid:18) d | Y | g (cid:100) TX − (cid:19) ω X . We repeat the constructions of Section 13, except that ω X is nowreplaced by ω Xθ . An extra subscript θ will be introduced to distinguishthe objects constructed here from the ones in Section 13. Instead of OHERENT SHEAVES AND RRG 145 (13.35), we have A (cid:48)(cid:48) Z = A (cid:48)(cid:48) + i z , A (cid:48) Z,θ = e q ∗ iω Xθ ( A (cid:48) + i z ) e − q ∗ iω Xθ , A Z,θ = A (cid:48)(cid:48) Z + A (cid:48) Z,θ . (16.3)The following result was established in [B13, Theorem 11.1.2]. Theorem 16.2. The following identities hold: A (cid:48) Z,θ = A (cid:48) Z + c (cid:18) d | Y | g (cid:100) TX − (cid:19) (cid:0) z ∗ − q ∗ ∂ X iω X (cid:1) + cdq ∗ iω X i (cid:98) y , (16.4) A Z,θ = A Z + c (cid:18) d | Y | g (cid:100) TX − (cid:19) (cid:0) z ∗ − q ∗ ∂ X iω X (cid:1) + cdq ∗ iω X i (cid:98) y . Proof. This is a consequence of (13.30), (13.37), and (16.2). (cid:3) As in (13.53), by replacing ω X by ω Xθ , we can define a new Hermitianform (cid:15) X,θ associated with ω Xθ . We obtain an analogue of Theorem 13.15.16.2. A formula for A Z,θ . We use the notation of Section 13.10. Inparticular (cid:98) w , . . . , (cid:98) w n denotes an orthonormal basis of (cid:16)(cid:100) T X, g (cid:100) T X (cid:17) . Let (cid:98) y ∗ ∈ (cid:91) T ∗ X be dual to (cid:98) y with respect to the metric g (cid:100) T X .Now, we give a form of [B13, Theorem 11.2.1]. Theorem 16.3. The following identity holds: (16.5) A Z,θ = A Z + c (cid:18) d | Y | g (cid:100) TX − (cid:19) (cid:18) | Z | g TX − i z q ∗ ∂ X iω X (cid:19) + cdz ∗ i (cid:98) y + c (cid:18) d | Y | g (cid:100) TX − (cid:19) (cid:16) ∇ T X (cid:48)(cid:48) ,H z ∗ − q ∗ ∂ X ∂ X iω X (cid:17) + cd (cid:16) q ∗ ∂ X iω X (cid:17) i (cid:98) y + c (cid:32) d (cid:98) y ∗ (cid:0) z ∗ − q ∗ ∂ X iω X (cid:1) + (cid:18) d | Y | g (cid:100) TX − (cid:19) (cid:98) w i ∇ (cid:98) w i z ∗ + dq ∗ iω X (cid:16) ∇ V (cid:98) y + (cid:98) w i i (cid:98) w i (cid:17)(cid:33) . Proof. Since our fibration is product, and the metric g T X is constantover S , the tensor σ defined in [B13, Definition 2.2.3] vanishes identi-cally. Using [B13, Theorem 11.2.1] and equation (13.70), we get (16.5).The extra terms in the right-hand side of (16.5) do not depend on (cid:0) E , A E (cid:48)(cid:48) (cid:1) , because C = B + B ∗ contains only exterior products. Theproof of our theorem is completed. (cid:3) 46 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI The superconnection A Y,θ . Here we take z = y . Using (13.83),(16.5), we get(16.6) A Y,θ = A Y + c (cid:18) d | Y | g (cid:100) TX − (cid:19) (cid:18) | Y | g TX − i y q ∗ ∂ X iω X (cid:19) + cdy ∗ i (cid:98) y + c (cid:18) d | Y | g (cid:100) TX − (cid:19) (cid:16) γy ∗ − q ∗ ∂ X ∂ X iω X (cid:17) + cd (cid:16) q ∗ ∂ X iω X (cid:17) i (cid:98) y + c (cid:18) d (cid:98) y ∗ (cid:0) y ∗ − q ∗ ∂ X iω X (cid:1) + (cid:18) d | Y | g (cid:100) TX − (cid:19) (cid:98) w i w i ∗ (cid:19) + cdq ∗ iω X (cid:16) ∇ V (cid:98) y + (cid:98) w i i (cid:98) w i (cid:17) . By (16.6), A Y,θ is fibrewise hypoelliptic.Set(16.7) V θ ( Y ) = (1 − c ) 12 | Y | g TX + cd | Y | g TX | Y | g (cid:100) TX . By (13.81), (13.82), and (16.6), the potential V θ appears in A Y,θ . Given d > 0, there exists C d > c ∈ [0 , V θ ( Y ) ≥ C d (cid:16) | Y | g TX − (cid:17) . The condition d > c = 1.The same arguments as in [B13, Section 11.3] and (16.6), (16.8) showthat exp (cid:0) −A Y,θ (cid:1) is fibrewise trace class.We use the same notation as in Subsection 14.1. Given p , let R p be the collection of parameters ω X , g D , g (cid:100) T X , θ with g (cid:100) T X ∈ p . Let d R p be the de Rham operator on R p . Then (cid:15) − X,θ d R p (cid:15) X,θ is a 1-form on R p with values in (cid:15) X,θ -self-adjoint endomorphisms. Definition 16.4. For θ = ( c, d ), c ∈ [0 , , d > 0, put(16.9) ch (cid:16) A (cid:48)(cid:48) Y , ω X , g D , g (cid:100) T X , θ (cid:17) = ϕ Tr s (cid:2) exp (cid:0) −A Y,θ (cid:1)(cid:3) . Theorem 16.5. The form ch (cid:16) A (cid:48)(cid:48) Y , ω X , g D , g (cid:100) T X , θ (cid:17) lies in Ω (=) ( S, R ) ,it is closed, and its Bott-Chern cohomology class does not depend on ω X , g D , g (cid:100) T X , θ , or on the splitting of E .Given p , ϕ Tr s (cid:2) (cid:15) − X,θ d R p (cid:15) X,θ exp (cid:0) −A Y,θ (cid:1)(cid:3) is a -form on R p with val-ues in Ω (=) ( S, R ) , and moreover, (16.10) d R p ch (cid:16) A (cid:48)(cid:48) Y , ω X , g D , g (cid:100) T X , θ (cid:17) = − ∂ S ∂ S iπ ϕ Tr s (cid:2) (cid:15) − X,θ d R p (cid:15) X,θ exp (cid:0) −A Y,θ (cid:1)(cid:3) . OHERENT SHEAVES AND RRG 147 The class of the forms ch (cid:16) A (cid:48)(cid:48) Y , ω X , g D , g (cid:100) T X , θ (cid:17) in Bott-Chern cohomol-ogy coincides with ch BC ( A (cid:48)(cid:48) Y ) .Proof. The proof follows the same lines as the proof of [B13, Theorem11.3.1] and of Theorem 14.2. The arguments of [B13] show that theydepend smoothly on all the parameters. Also for c = 0, by (16.4), weget(16.11) ch (cid:16) A (cid:48)(cid:48) Y , ω X , g D , g (cid:100) T X , θ (cid:17) = ch (cid:16) A (cid:48)(cid:48) Y , ω X , g D , g (cid:100) T X (cid:17) , which gives the last statement in our theorem, and concludes its proof. (cid:3) The scaling identities. For c ∈ [0 , , d > , t > 0, put(16.12) θ t = ( c, dt ) . As in (15.20), by (16.4), we have a splitting(16.13) A Y,θ = A Y,θ + C. Let A Y,θ,t be the superconnection A Y,θ associated with ω X /t, g D , g (cid:100) T X /t , ω X /t, θ. We have an analogue of [B13, eq. (11.3.12)] and of Proposition 15.5. Proposition 16.6. The following identity holds: (16.14) t N (cid:98) V / N V / K t A Y,θ t ,t K − t t − N (cid:98) V / − N V / = t − N H / (cid:16) √ t A Y,θ + t N/ Ct − N/ (cid:17) t N H / . Proof. By combining Proposition 15.5 and Theorem 16.2, we get (16.14).The proof of our proposition is completed. (cid:3) Let A tY,θ be the superconnection A Y,θ associated with ω X /t, g D , g (cid:100) T X , θ .By combining (15.24) and (16.4), we get(16.15) A tt / Y,θ = δ ∗ t / A Y,θ,t δ ∗− t / . By (16.14), (16.15), we get the analogue of [B13, eq. (11.5.2)],(15.25),(16.16) K / √ t t N V / A tt / Y,θ t t − N V / K √ t = t − N H / (cid:16) √ t A Y,θ + t N/ Ct − N/ (cid:17) t N H / . We use the notation,(16.17) M θ,t = A t, t / Y,θ . 48 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI Recall that M t was defined in (15.26) and is given by (15.27). Using(16.5), we get(16.18) M θ,t = M t + c (cid:18) d | Y | g (cid:100) TX − (cid:19) (cid:18) t | Y | g TX − t / i y q ∗ ∂ X iω X (cid:19) + cdt / y ∗ i (cid:98) y + c (cid:18) d | Y | g (cid:100) TX − (cid:19) (cid:16) t / γy ∗ − q ∗ ∂ X ∂ X iω X /t (cid:17) + cd (cid:16) q ∗ ∂ X iω X (cid:17) i (cid:98) y /t + c (cid:18) d (cid:98) y ∗ (cid:0) t / y ∗ − q ∗ ∂ X iω X /t (cid:1) + (cid:18) d | Y | g (cid:100) TX − (cid:19) t / (cid:98) w i w i ∗ (cid:19) + cdt q ∗ iω X (cid:16) ∇ V (cid:98) y + (cid:98) w i i (cid:98) w i (cid:17) . The uniform estimates. In the sequel, we will make c = 1, sothat for d > , t > θ t = (1 , dt ) . Put(16.20) M Y,θ,t = K ( dt ) − / (cid:16) √ t A Y,θ + t N/ Ct − N/ (cid:17) K ( dt ) / . Recall that d ( x, x (cid:48) ) denotes the Riemannian distance on X withrespect to (cid:98) g T X . Here is an important result, which is an analogue of[B13, Theorem 11.5.1]. Theorem 16.7. Given k > , there exist m ∈ N , c > , C > suchthat for t ∈ ]0 , , d ∈ [ kt , , z = ( x, Y ) , z (cid:48) = ( x (cid:48) , Y (cid:48) ) ∈ X , we have (16.21) | exp ( −M Y,θ,t ) ( z, z (cid:48) ) |≤ Ct m exp (cid:16) − c (cid:16) | Y | g (cid:100) TX + | Y (cid:48) | g (cid:100) TX + (cid:0) d/t (cid:1) / d ( x, x (cid:48) ) (cid:17)(cid:17) . Proof. If we make C = 0 , the proof of our theorem was already givenin [B13, Theorem 11.5.1]. Let us explain why the introduction of theextra term containing t N/ Ct − N/ in (16.20) does not change anythingto the estimates. Indeed this term remains uniformly bounded. Whentaking the square in the right-hand side of (16.20), this introduces,after scaling, a term which is of the form ( t/d ) / i Y t N/ Ct − N/ . In C ,the component of total degree 0 in Λ ( T ∗ C X ) is necessarily disregarded,which introduces an extra factor which is at least √ t . The factor thatappears is then of the order at most ( t /d ) / Y . Under our assumptions As the reader may have guessed, here C refers to the tensor defined in (7.21),and not to the constant appearing in our theorem. OHERENT SHEAVES AND RRG 149 on d , t /d remains uniformly bounded, so that this term is of order | Y | .Under the indicated rescaling, a potential of the order of | Y | appears in K ( dt ) − / t A Y,θ K ( dt ) / . The term | Y | being much bigger than | Y | , thereis no difficulty in extending these estimates to the present situation. (cid:3) Remark . Using (16.16) and (16.20), Theorem 16.7 will play therole of Proposition 15.6.16.6. The exotic superconnection forms of vector bundles. Let Z be a complex manifold, and let F be a holomorphic vector bundleon Z .Let g F be a Hermitian metric on F , let ω Z be a smooth real (1 , Z . In [B13, Definition 10.2.1], hypoelliptic superconnectionforms a (cid:0) F, ω Z , g F (cid:1) on Z were defined, that have the following prop-erties: • They depend smoothly on the parameters. • They lie in Ω (=) ( Z, R ), they are closed, and their Bott-Cherncohomology class does not depend on the parameters. • They are unchanged by constant rescaling of g F . • The following identity holds:(16.22) a (cid:0) F, , g F (cid:1) = Td (cid:0) F, g F (cid:1) . If a BC ( F ) is the common Bott-Chern class of the forms a (cid:0) F, ω Z , g F (cid:1) ,then(16.23) a BC ( F ) = Td BC ( F ) . The limit as t → of the forms ch (cid:16) A (cid:48)(cid:48) Y , ω X /t, g D , g (cid:100) T X /t , θ t (cid:17) . Using (15.27), (16.18), we obtain a formula closely related to [B13, eq. With the notation of [B13], these forms are instead denoted a g (cid:16) b, d, g (cid:98) F (cid:17) . Here,we make g = 1 , b = 1. Also d stands for a factor multiplying ω Z . 50 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI (11.5.1)],(16.24) M θ t ,t = − 12 ∆ Vg (cid:100) TX + dt | Y | g TX | Y | g (cid:100) TX + t / (cid:16) (cid:98) w i ∧ i w i − i (cid:98) w i i w i (cid:17) + t / (cid:98) ∇ F t,Y + t / i Y C − ∇ VR (cid:100) TX (cid:98) Y − (cid:68) R (cid:100) T X (cid:98) w i , (cid:98) w j (cid:69) (cid:98) w i i (cid:98) w j + dt / | Y | g (cid:100) TX (cid:98) w i w i ∗ − | Y | g (cid:100) TX q ∗ ∂ X ∂ X idω X + t / (cid:18) dt | Y | g (cid:100) TX − (cid:19) (cid:0) γy ∗ − i y q ∗ ∂ X iω X (cid:1) + dt / (cid:16) y ∗ i (cid:98) y + (cid:98) y ∗ y ∗ (cid:17) + (cid:16) q ∗ ∂ X idω X (cid:17) i (cid:98) y − (cid:98) y ∗ q ∗ ∂ X idω X + q ∗ idω X (cid:16) ∇ V (cid:98) y + (cid:98) w i i (cid:98) w i (cid:17) + A E , . Given ρ ∈ ]0 , θ t = (cid:0) , dt ρ +1 (cid:1) . We establish an extension of [B13, Theorem 11.6.1]. We use thenotation of Subsection 16.6, with Z = X , F = T X , ω Z = ω X , g F = (cid:98) g T X . Theorem 16.9. As t → , then ch (cid:16) A (cid:48)(cid:48) Y , ω X /t, g D , g (cid:100) T X /t , θ t (cid:17) → p ∗ (cid:2) q ∗ a (cid:0) T X, dω X , (cid:98) g T X (cid:1) ch (cid:0) A E (cid:48)(cid:48) , g D (cid:1)(cid:3) , (16.26)ch (cid:16) A (cid:48)(cid:48) Y , ω X /t, g D , g (cid:100) T X /t , θ t (cid:17) → p ∗ (cid:2) q ∗ Td (cid:0) T X, (cid:98) g T X (cid:1) ch (cid:0) A E (cid:48)(cid:48) , g D (cid:1)(cid:3) . Proof. By (16.9), (16.15), and (16.17), we get(16.27) ch (cid:16) A (cid:48)(cid:48) Y , ω X /t, g D , g (cid:100) T X /t , θ (cid:17) = ϕ Tr s [exp ( − M θ,t )] . Let P θ,t (( x, Y ) , ( x (cid:48) , Y (cid:48) )) be the smooth kernel for exp ( − M θ,t ) withrespect to dx (cid:48) dY (cid:48) (2 π ) n . By (16.27), as in (15.37), we get(16.28) ch (cid:16) A (cid:48)(cid:48) Y , ω X /t, g D , g (cid:100) T X /t , θ (cid:17) = ϕ (cid:90) X (cid:34)(cid:90) (cid:92) T R ,x X Tr s [ P θ,t (( x, Y ) , ( x, Y ))] dY (2 π ) n (cid:35) (cid:99) dx (2 π ) n . Put(16.29) m θ,t ( x ) = ϕ π ) n (cid:90) (cid:92) T R ,x X Tr s [ P θ,t (( x, Y ) , ( x, Y ))] dY (2 π ) n , OHERENT SHEAVES AND RRG 151 so that(16.30) ch (cid:16) A (cid:48)(cid:48) Y , ω X /t, g D , g (cid:100) T X /t , θ (cid:17) = (cid:90) X m θ,t ( x ) (cid:99) dx. We fix d > 0. In (16.30), we replace θ by θ t . As explained before,given x ∈ X , Theorem 16.7 takes care of localization as t → 0, andthis uniformly in ( s, x ) ∈ M . If instead θ t is replaced by θ t , i.e., wesubstitute d by dt ρ in θ , since ρ ∈ ]0 , θ t , or θ t . Weconstruct an operator N θ,t,x on H x exactly as in the proof of Theo-rem 15.7, that coincides with M θ,t on B T R ,x X (0 , (cid:15)/ × (cid:92) T R ,x X . Let Q θ,t,x (( U, Y ) , ( U (cid:48) , Y (cid:48) )) be the smooth kernel for exp ( − N θ,t,x ) with re-spect to the volume dU (cid:48) dY (cid:48) (2 π ) n . When studying the asymptotics of (16.30)as t → 0, we may as well replace m θ,t ( x ) by(16.31) n θ,t ( x ) = ϕ π ) n (cid:90) (cid:92) T R ,x X Tr s [ Q θ,t,x ((0 , Y ) , (0 , Y ))] dY (2 π ) n . Recall that P ,s,x is given by (15.52). Given d ≥ 0, put(16.32) P ,d,s,x = P ,s,x − | Y | g (cid:100) TX q ∗ ∂ X ∂ X idω X − (cid:98) y ∗ q ∗ ∂ X idω X + (cid:16) q ∗ ∂ X idω X (cid:17) i (cid:98) y + q ∗ idω X (cid:16) ∇ (cid:98) y + (cid:98) w i i (cid:98) w i (cid:17) , the tensors in (16.32) being evaluated at ( s, x ).We claim that as t → 0, instead of (15.53), we have P θ t ,t,x → P ,d,s,x , P θ t ,t,x → P ,s,x . (16.33)To establish (16.33), we will refer to the proof of (15.53), and also toequation (16.18), with c = 1, and d replaced by dt .In the proof of (15.53), we made essential use of the fact ∂ X ∂ X iω X =0. In the sequel, we will view P ,s,x as the contribution to the limit ofthe terms in M t in (15.27), with the exception of the term − q ∗ ∂ X ∂ X ω X /t .However, in equation (16.18) where we replace θ by θ t , so that c = 1,this term goes away. Since the other terms in (16.18) do not con-tain operators like i w i , i w i , they are unaffected by the conjugation byexp (cid:0) − o /t / (cid:1) . Therefore, while replacing d by dt in the right-hand sideof (16.18), we can take their naive limit as t → 0, and we get (16.33).We can instead use equation (16.24) for M θ t ,t and obtain (16.33). 52 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI Let S θ t ,t,x (( U, Y ) , ( U (cid:48) , Y (cid:48) )) be the smooth kernel associated withexp ( − P θ t ,t ) with respect to the volume dU (cid:48) dY (cid:48) (2 π ) n . We claim that we canuse the same arguments as in [B13, eq. (11.6.9)] to control | S θ t ,t,x ((0 , Y ) , (0 , Y )) | when t ∈ ]0 , M θ t ,t withthe corresponding equation [B13, eq. (11.5.1)], we see that the maindifference lies in the presence of the term t / i Y C . However note thatgiven c > 0, there is c (cid:48) > a ∈ R ,(16.34) | a | ≤ c (cid:48) + ca . By (16.34), we deduce that(16.35) (cid:12)(cid:12) t / Y (cid:12)(cid:12) g (cid:100) TX ≤ c (cid:48) + ct | Y | g (cid:100) TX . By (16.35), we deduce that given ρ ∈ [0 , , d > 0, if d ≥ d t ρ , then(16.36) (cid:12)(cid:12) t / Y (cid:12)(cid:12) g (cid:100) TX ≤ c (cid:48) + cdt | Y | g (cid:100) TX . Using (16.24), (16.36), the same arguments as in [B13, eq. (11.6.9)]show that for d > , ρ ∈ [0 , c > , C > t ∈ ]0 , , ≥ d ≥ d t ρ , x ∈ X, Y ∈ (cid:92) T R ,x X , (16.37) | S θ t ,t,x ((0 , Y ) , (0 , Y )) | ≤ C exp (cid:16) − cd | Y | − ρ/ g (cid:100) TX (cid:17) . The estimate (16.37) also gives a corresponding estimate for S θ t ,t ((0 , Y ) , (0 , Y )).Recall that the kernel S ,s,x (( U, Y ) , ( U (cid:48) , Y (cid:48) )) for exp ( − P ,s,x ) wasdefined in the proof of Theorem 15.7. Let S ,d,s,x (( U, Y ) , ( U (cid:48) , Y (cid:48) )) bethe smooth kernel associated with exp ( − P ,d,s,x ) with respect to thevolume dU (cid:48) dY (cid:48) (2 π ) n . Using the uniform estimates (16.37) and proceeding asin the proof of [B13, Theorem 11.6.1], we find that for ρ ∈ ]0 , t → S θ t ,t,x (( U, Y ) , ( U (cid:48) , Y (cid:48) )) → S ,d,s,x (( U, Y ) , ( U (cid:48) , Y (cid:48) )) , (16.38) S θ t ,t,x (( U, Y ) , ( U (cid:48) , Y (cid:48) )) → S ,s,x (( U, Y ) , ( U (cid:48) , Y (cid:48) )) . In [B13], the condition d ≤ OHERENT SHEAVES AND RRG 153 By proceeding as in the proof of (15.65), using (16.38), we find thatas t → n θ t ,t ( x ) → n ,d,s ( x )= ϕ π ) n (cid:90) (cid:92) T R ,x X Tr sΛ (cid:18) (cid:91) T ∗ x X (cid:19)(cid:98) ⊗ D s,x [[ S ,d,s,x ((0 , Y ) , (0 , Y )) 1] max ] dY (2 π ) n , (16.39) n θ t ,t ( x ) → n ,s ( x )= ϕ π ) n (cid:90) (cid:92) T R ,x X Tr sΛ (cid:18) (cid:91) T ∗ x X (cid:19)(cid:98) ⊗ D s,x [[ S ,s,x ((0 , Y ) , (0 , Y )) 1] max ] dY (2 π ) n . By (16.37), given ρ ∈ ]0 , C > x ∈ X , | n θ t ,t ( x ) | ≤ C, (cid:12)(cid:12) n θ t ,t ( x ) (cid:12)(cid:12) ≤ C. (16.40)By (16.30), (16.39), and (16.40), as t → (cid:16) A (cid:48)(cid:48) Y , ω X /t, g D , g (cid:100) T X /t , θ t (cid:17) → (cid:90) X n ,d,s ( x ) (cid:99) dx, (16.41) ch (cid:16) A (cid:48)(cid:48) Y , ω X /t, g D , g (cid:100) T X /t , θ t (cid:17) → (cid:90) X n ,s ( x ) (cid:99) dx, By (15.73), (16.41), we get the second equation in (16.26).To compute the right-hand side of the first equation in (16.41), weproceed as in the proof of [B13, Theorem 11.6.1]. Consider equations(15.52), (16.32) that give a formula for P ,d,s,x . When A E is the trivialsuperconnection on C , this is exactly the operator obtained in [B13,eq. (11.6.5)]. As explained in detail in [B13], in this case, if we replaceformally the variables w i , w i by variables i w i , i w i that act on an extracopy of Λ (cid:0) T ∗ C ,x X (cid:1) , the operator P ,d,s,x coincides with the operator L d,Y considered in [B13, eq. (10.2.4)] that is associated with thevector bundle T X , the metric (cid:98) g T X , the (1 , ω X , and the section Y . When A E is non trivial, in P ,d,s,x , we have the extra term A E , s,x .Using the sign conventions explained in [B13, eq. (10.2.2)], thatmatch (15.68), and using the notation in Subsection 16.6, for d > (cid:90) X n ,d ( x ) (cid:99) dx = p ∗ (cid:2) q ∗ a (cid:0) T X, dω X , (cid:98) g T X (cid:1) ch (cid:0) A E (cid:48)(cid:48) , g D (cid:1)(cid:3) By (16.41), (16.42), we obtain the first equation in (16.26). The proofof our theorem is completed. (cid:3) In [B13], because of the presence of an extra parameter b > 0, this operator isdenoted L d,Y b . 54 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI Theorem 16.10. The following identity holds: (16.43) ch BC ( A (cid:48)(cid:48) Y ) = p ∗ (cid:2) q ∗ Td BC ( T X ) ch BC (cid:0) A E (cid:48)(cid:48) (cid:1)(cid:3) in H (=)BC ( S, R ) . Proof. Using Theorem 16.5 and either (16.23) combined with the firstequation in (16.26), or the second row in (16.26), we get (16.43). Theproof of our theorem is completed. (cid:3) A proof of Theorem 10.1. By combining Theorems 14.5 and16.10, we get Theorem 10.1, when F = E . As explained in Subsection10.2, this is enough to establish Theorem 10.1 in full generality. Thisalso completes the proof of our main result stated in Theorem 1.1. ndex A (cid:48)(cid:48) , 115 (cid:98) A ( B ), 15 α ∗ , 23[ α ], 10 { α } , 10, 110[ A, B ] + , 10[ a, b ], 9, 21 A D(cid:48)(cid:48) , 87 A D , 90 A D(cid:48)(cid:48)∗ , 90 A E (cid:48)(cid:48) , 28 A E (cid:48) , 59 A E , 60 A E M (cid:48)(cid:48) z , 117 A E M (cid:48)(cid:48) z , 117 A E (cid:48) T , 67 A (cid:48) , 116 A p ∗ E , 88, 90, 127 A p ∗ E (cid:48) , 89 A p ∗ E (cid:48)(cid:48) , 87 A E , t , 136 A tY,θ , 147 A tY , 135Aut ( E ), 20Aut ( E ), 20 A (cid:48) Y,b , 126 A Y,b , 126 A Y,θ,t , 147 A Y,t , 134 A (cid:48) Z , 117 A (cid:48)(cid:48) Z , 117 A Z , 117 B , 29, 59 β , 90B( X ), 42B( X ), 42 B (cid:48) Z , 118 B (cid:48)(cid:48) Z , 118 B Z , 118 C , 60, 117 c ( B ), 15 c , 89ch BC ( A (cid:48)(cid:48) Y ), 130 c B , 25 C bcoh ( X ), 16ch BC (cid:16) A E (cid:48)(cid:48) (cid:17) , 65ch BC (cid:0) A p ∗ E (cid:48)(cid:48) (cid:1) , 98, 130ch BC ( E ), 71ch BC ( F ), 71C b ( X ), 16ch ( B ), 15ch (cid:0) A E (cid:48)(cid:48) , h (cid:1) , 61ch (cid:0) A p ∗ E (cid:48)(cid:48) , ω X , g D (cid:1) , 97, 130ch (cid:0) HD, h HD (cid:1) , 68coh ( X ), 16 c t , 105 C W , 19 C X , 30 C X , 30 C X , 112 C Y,b , 126 D , 19 D , 87, 127 δ , 127 D a, ± , 99 δ a , 126 δ ∗ a , 126D bcoh ( X ), 16D b (X), 16det D , 35deg, 23deg − , 23, 57 d F ∞ , 41det H E , 35 d M D , 61 ∂ V , 113 dv X , 119 D X, LC , 94 (cid:99) dx , 136 d ( x, x (cid:48) ), 136, 148 D ( X, C ), 11 D ( X, R ), 11 D Y,b , 126, 128 E , 118, 127 E , 33 E , 19, 28 E ,a, ± , 74 BISMUT , SHU SHEN , AND ZHAOTING WEI E (cid:98) ⊗ b F , 40 E (cid:98) ⊗ b F , 40 E † , 19 E M , 114 E M , 115 E M , 114 (cid:15) M , 120 (cid:15) M,b , 126End ( E ), 20 η X , 119 E X , 42 (cid:15) X , 120 (cid:15) X,b , 126 η X,θ , 145 F , 118, 122 F , 123 F ∞ , 41 F ∞ , 41 ϕ , 61 f ! , 16 f ! , 17 f ∗ b , 39 F X , 43 F X , 52 γ , 125 g † h , 24, 63 (cid:98) g T X , 121 g T X , 88, 117 H p,q BC ( X, C ), 10 HD , 33 h det D , 59, 68 H E , 33 H HD , 34 (cid:98) ⊗ , 7 HQ , 22 h T , 67 H (=)BC ( X, C ), 10 H (=)BC ( X, R ), 10 H • ( X, R ), 9 I , 112 I c , 113 i T , 25 K a , 126, 134 K (cid:0) D bcoh ( X ) (cid:1) , 71 K ( X ), 16, 71 K X , 94 K · ( X ), 4 K · ( X ), 4 (cid:98) L , 127 (cid:98) Λ, 127 Lf ∗ , 16 M (cid:48) t , 135 M (cid:48)(cid:48) t , 136 M t , 135 N D , 62, 67 ∇ D (cid:48)(cid:48) ,X ∗ α , 89 ∇ D (cid:48) , 59, 89 ∇ D (cid:48) , 59 ∇ D , 60 ∇ det D (cid:48)(cid:48) , 35 ∇ D (cid:98) ⊗ q ∗ I (cid:48)(cid:48) , 115 ∇ D (cid:98) ⊗ q ∗ I (cid:48) , 116 ∇ D,S , 90 ∇ F , 92 (cid:98) ∇ F ,a , 122 (cid:98) ∇ F ,a a , 123 (cid:98) ∇ F , 123 (cid:98) ∇ F , 123 ∇ F , 125 (cid:98) ∇ F , 136 (cid:98) ∇ F t , 135 (cid:98) ∇ F t , 135 ∇ F t , 136 (cid:98) ∇ F t , 136 ∇ HD (cid:48)(cid:48) , 33 ∇ Λ ( T ∗ X ) (cid:98) ⊗ D, LC , 94 ∇ Λ ( T ∗ X ), 90 ∇ Λ ( T ∗ X ) (cid:98) ⊗ D , 94 ∇ Λ ( T ∗ X ), 93 (cid:98) ∇ Λ( T ∗ R X ) ,a , 122 ∇ Λ( T ∗ Z )a , 25 N M , 114 ∇ T R X, LC , 90 ∇ T R X , 91 ∇ T R X , 91 ∇ T X , 90 ∇ (cid:100) T X , 112 ∇ T X , 91 OHERENT SHEAVES AND RRG 157 (cid:98) ∇ T X,a , 122 (cid:98) ∇ T X , 121 ∇ T Z,a , 26 N X , 112 N X , 112 o , 139 (cid:98) ⊗ b , 115 ω Xθ , 144 (cid:98) ω X ,V , 127Ω − ,i ( X, C ), 57 ω X , 88, 117Ω (=) ( X, C ), 10Ω − ,i ( X, D ), 57 O ∞ X ( F ), 33Ω (=) ( X, R ), 10 P , 28, 127 P , 97 p , 133 P a, − , 74, 99 P BC ( F ), 15 p ∗ E ,a, ± , 100 P (cid:0) F, g F (cid:1) , 13 P ( F ), 13 P ,s,x , 140, 151 P ,d,x , 151 P (cid:48)(cid:48) t ( z, z (cid:48) ), 136 P t ( z, z (cid:48) ), 136 Q p , 129 θ t , 150 Rf ∗ , 17 RH F , 72, 101 R p , 146 R T R X , 91 R T R X , 91 R X , 48 σ , 119 σ ∗ , 119 σ ∗ , 119 S ,s,x (( U, Y ) , ( U (cid:48) , Y (cid:48) )), 142, 152 τ , 90 θ t , 147 (cid:98) τ , 121Td ( B ), 15 θ , 58, 144 θ h , 58 ϑ , 90 T H X , 112 (cid:101) , 23, 57Tr s , 21 (cid:100) T X , 111 U H , 113 v , 29 X , 111 (cid:98) y , 113 y , 124 Z , 117 Z − , 121 z − , 121 Z Hom ( E , E ) X , 42, 52 58 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI References [BGV92] N. Berline, E. Getzler, and M. Vergne. Heat kernels and Dirac operators .Grundl. Math. Wiss. Band 298. Springer-Verlag, Berlin, 1992.[B86] J.-M. Bismut. The Atiyah-Singer index theorem for families of Dirac op-erators: two heat equation proofs. Invent. Math. , 83(1):91–151, 1986.[B89b] J.-M. Bismut. A local index theorem for non-K¨ahler manifolds. Math.Ann. , 284(4):681–699, 1989.[B90] J.-M. Bismut. Koszul complexes, harmonic oscillators, and the Toddclass. J. Amer. Math. Soc. , 3(1):159–256, 1990. With an appendix bythe author and C. Soul´e.[B97] J.-M. Bismut. Holomorphic families of immersions and higher analytictorsion forms. Ast´erisque , (244):viii+275, 1997.[B05] J.-M. Bismut. The hypoelliptic Laplacian on the cotangent bundle. J.Amer. Math. Soc. , 18(2):379–476 (electronic), 2005.[B08] J.-M. Bismut. The hypoelliptic Dirac operator. In Geometry and dynam-ics of groups and spaces , volume 265 of Progr. Math. , pages 113–246.Birkh¨auser, Basel, 2008.[B12] J.-M. Bismut. Index theory and the hypoelliptic Laplacian. In Metricand differential geometry , number 297 in Progress in Mathematics, pages181–232. Birkh¨auser/Springer, Basel, 2012.[B13] J.-M. Bismut. Hypoelliptic Laplacian and Bott-Chern cohomology , volume305 of Progress in Mathematics . Birkh¨auser/Springer, Cham, 2013. Atheorem of Riemann-Roch-Grothendieck in complex geometry.[BF86] J.-M. Bismut and D.S. Freed. The analysis of elliptic families. II. Dirac op-erators, eta invariants, and the holonomy theorem. Comm. Math. Phys. ,107(1):103–163, 1986.[BGS88a] J.-M. Bismut, H. Gillet, and C. Soul´e. Analytic torsion and holomorphicdeterminant bundles. I. Bott-Chern forms and analytic torsion. Comm.Math. Phys. , 115(1):49–78, 1988.[BGS88b] J.-M. Bismut, H. Gillet, and C. Soul´e. Analytic torsion and holomorphicdeterminant bundles. II. Direct images and Bott-Chern forms. Comm.Math. Phys. , 115(1):79–126, 1988.[BGS88c] J.-M. Bismut, H. Gillet, and C. Soul´e. Analytic torsion and holomorphicdeterminant bundles. III. Quillen metrics on holomorphic determinants. Comm. Math. Phys. , 115(2):301–351, 1988.[BGS90a] J.-M. Bismut, H. Gillet, and C. Soul´e. Bott-Chern currents and compleximmersions. Duke Math. J. , 60(1):255–284, 1990.[BGS90b] J.-M. Bismut, H. Gillet, and C. Soul´e. Complex immersions and Arakelovgeometry. In The Grothendieck Festschrift, Vol. I , volume 86 of Progr.Math. , pages 249–331. Birkh¨auser Boston, Boston, MA, 1990.[BL08] J.-M. Bismut and G. Lebeau. The hypoelliptic Laplacian and Ray-Singermetrics , volume 167 of Annals of Mathematics Studies . Princeton Univer-sity Press, Princeton, NJ, 2008.[BL95] J.-M. Bismut and J. Lott. Flat vector bundles, direct images and higherreal analytic torsion. J. Amer. Math. Soc. , 8(2):291–363, 1995.[Bl10] J. Block. Duality and equivalence of module categories in noncommuta-tive geometry. In A celebration of the mathematical legacy of Raoul Bott , OHERENT SHEAVES AND RRG 159 volume 50 of CRM Proc. Lecture Notes , pages 311–339. Amer. Math. Soc.,Providence, RI, 2010.[Bor87] A. Borel, P.-P. Grivel, B. Kaup, A. Haefliger, B. Malgrange, and F. Ehlers. Algebraic D -modules , volume 2 of Perspectives in Mathematics . AcademicPress, Inc., Boston, MA, 1987.[BorS58] A. Borel and J.-P. Serre. Le th´eor`eme de Riemann-Roch. Bull. Soc. Math.France , 86:97–136, 1958.[BoC65] R. Bott and S. S. Chern. Hermitian vector bundles and the equidistribu-tion of the zeroes of their holomorphic sections. Acta Math. , 114:71–112,1965.[Col71] Collective. Th´eorie des intersections et th´eor`eme de Riemann-Roch . Lec-ture Notes in Mathematics, Vol. 225. Springer-Verlag, Berlin-New York,1971. S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie 1966–1967 (SGA6), Dirig´e par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collabo-ration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaudet J. P. Serre.[D09] J.-P. Demailly. Complex analytic and differential geometry Geometry of four-manifolds . ICM Series. AmericanMathematical Society, Providence, RI, 1988. A plenary address presentedat the International Congress of Mathematicians held in Berkeley, Cali-fornia, August 1986, Introduced by I. M. James.[G86] E. Getzler. A short proof of the local Atiyah-Singer index theorem. Topol-ogy , 25(1):111–117, 1986.[GlJ87] J. Glimm and A. Jaffe. Quantum physics . Springer-Verlag, New York,second edition, 1987. A functional integral point of view.[GrR84] H. Grauert and R. Remmert. Coherent analytic sheaves , volume 265 of Grundlehren der Mathematischen Wissenschaften [Fundamental Princi-ples of Mathematical Sciences] . Springer-Verlag, Berlin, 1984.[Gr60] H. Grauert. Ein Theorem der analytischen Garbentheorie und die Mod-ulr¨aume komplexer Strukturen. Inst. Hautes ´Etudes Sci. Publ. Math. ,(5):64, 1960.[Gre80] H. I. Green. Chern classes for coherent sheaves . PhD thesis, Universityof Warwick, January 1980.[Gri10] J. Grivaux. Chern classes in Deligne cohomology for coherent analyticsheaves. Math. Ann. , 347(2):249–284, 2010.[H67] L. H¨ormander. Hypoelliptic second order differential equations. ActaMath. , 119:147–171, 1967.[H83] L. H¨ormander. The analysis of linear partial differential operators. I , vol-ume 256 of Grundlehren der Mathematischen Wissenschaften [Fundamen-tal Principles of Mathematical Sciences] . Springer-Verlag, Berlin, 1983.Distribution theory and Fourier analysis.[Ho20a] T. Hosgood. Simplicial Chern-Weil theory for coherent analytic sheaves,part I. arXiv e-prints , page arXiv:2003.10023, March 2020.[Ho20b] T. Hosgood. Simplicial Chern-Weil theory for coherent analytic sheaves,part II. arXiv e-prints , page arXiv:2003.10591, March 2020. 60 JEAN-MICHEL BISMUT , SHU SHEN , AND ZHAOTING WEI [KaS90] M. Kashiwara and P. Schapira. Sheaves on manifolds , volume 292 of Grundlehren der Mathematischen Wissenschaften [Fundamental Princi-ples of Mathematical Sciences] . Springer-Verlag, Berlin, 1990. With achapter in French by Christian Houzel.[Ke06] B. Keller. On differential graded categories. In International Congress ofMathematicians. Vol. II , pages 151–190. Eur. Math. Soc., Z¨urich, 2006.[KnM76] F. F. Knudsen and D. Mumford. The projectivity of the moduli spaceof stable curves. I. Preliminaries on “det” and “Div”. Math. Scand. ,39(1):19–55, 1976.[KN69] S. Kobayashi and K. Nomizu. Foundations of differential geometry. Vol.II . Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol.II. Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969.[KoS60] K. Kodaira and D. C. Spencer. On deformations of complex analyticstructures. III. Stability theorems for complex structures. Ann. of Math.(2) , 71:43–76, 1960.[Ma67] B. Malgrange. Ideals of differentiable functions . Tata Institute of Funda-mental Research Studies in Mathematics, No. 3. Tata Institute of Funda-mental Research, Bombay; Oxford University Press, London, 1967.[MQu86] V. Mathai and D. Quillen. Superconnections, Thom classes, and equivari-ant differential forms. Topology , 25(1):85–110, 1986.[Q16] H. Qiang. On the Bott-Chern characteristic classes for coherent sheaves. arXiv e-prints , page arXiv:1611.04238, November 2016.[Q17] H. Qiang. Bott-Chern characteristic forms and index theorems for coher-ent sheaves on complex manifolds . PhD thesis, University of Pennsylvania,2017.[Qu85] D. Quillen. Superconnections and the Chern character. Topology ,24(1):89–95, 1985.[Sc07] M. Schweitzer. Autour de la cohomologie de Bott-Chern. arXiv e-prints ,page arXiv:0709.3528, September 2007.[St20] The Stacks Project Authors. The stacks project.https://stacks.math.columbia.edu, 2020.[TT86] D. Toledo and Y. L. L. Tong. Green’s theory of Chern classes and theRiemann-Roch formula. In The Lefschetz centennial conference, Part I(Mexico City, 1984) , volume 58 of Contemp. Math. , pages 261–275. Amer.Math. Soc., Providence, RI, 1986.[V02] C. Voisin. A counterexample to the Hodge conjecture extended to K¨ahlervarieties. Int. Math. Res. Not. , (20):1057–1075, 2002.[W20] X. Wu. Intersection theory and Chern classes in Bott-Chern cohomology. arXiv e-prints , page arXiv:2011.13759, November 2020. OHERENT SHEAVES AND RRG 161 Institut de Math´ematique d’Orsay, Universit´e Paris-Sud, Bˆatiment307, 91405 Orsay, France Email address : [email protected] Institut de Math´ematiques de Jussieu-Paris Rive-Gauche, SorbonneUniversit´e, Case Courrier 247, 4 place Jussieu, 75252 Paris Cedex 05,France Email address : [email protected] Department of Mathematics, Texas A&M University-Commerce, Com-merce, TX 75429, USA Email address ::