Heat kernel asymptotics, local index theorem and trace integrals for CR manifolds with S 1 action
aa r X i v : . [ m a t h . DG ] J u l HEAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALSFOR CR MANIFOLDS WITH S ACTION
JIH-HSIN CHENG, CHIN-YU HSIAO, AND I-HSUN TSAIA
BSTRACT . Among those transversally elliptic operators initiated by Atiyah and Singer, Kohn’s (cid:3) b opera-tor on CR manifolds with S action is a natural one of geometric significance for complex analysts. Ourfirst main result establishes an asymptotic expansion for the heat kernel of such an operator with valuesin its Fourier components, which involves an unprecedented contribution in terms of a distance functionfrom lower dimensional strata of the S -action. Our second main result computes a local index density,in terms of tangential characteristic forms, on such manifolds including Sasakian manifolds of interest inString Theory, by showing that certain non-trivial contributions from strata in the heat kernel expansionwill eventually cancel out by applying Getzler’s rescaling technique to off-diagonal estimates. This leadsto a local result which can be thought of as a type of local index theorem on these CR manifolds. As ap-plications of our CR index theorem we can prove a CR version of Grauert-Riemenschneider criterion, andproduce many CR functions on a weakly pseudoconvex CR manifold with transversal S action and manyCR sections on some class of CR manifolds, answering (on this class of manifolds) some long-standingquestions in several complex variables and CR geometry. We give examples of these CR manifolds, someof which arise from Brieskorn manifolds. Moreover in some cases, without use of equivariant cohomol-ogy method nor keeping contributions arising from lower dimensional strata as done in previous works,we can reinterpret Kawasaki’s Hirzebruch-Riemann-Roch formula for a complex orbifold with an orbifoldholomorphic line bundle, as an index theorem obtained by a single integral over a smooth CR manifoldwhich is essentially the circle bundle of this line bundle. By contrast, if one computes the trace integral(instead of supertrace as in the case of index theorems) of our heat kernel, then the contributions arisingfrom the stratification of the S action necessarily occur in a nontrivial way. Some explicit expressionsabout these corrections are obtained in this paper.In short, besides certain applications our paper treats three major topics: i) an asymptotic expansion ofa (transversal) heat kernel related to Kohn Laplacian (Theorem 1.3); ii) a formulation of a local CR indextheorem for the case of locally free S action (Corollary 1.13); iii) study of this heat kernel trace integralin terms of some explicit invariants as reflections upon the geometry of the S stratification inside the CRmanifold (Theorems 1.14, 7.20 and 7.24). Among the three topics, the first topic is foundational. Thethird topic focuses on the role of the Gaussian part of the heat kernel (which is boiled down to a Diractype delta function on the S stratification) while the second topic does mainly on the non-Gaussian part.Jointly, the three topics explore and integrate the separate aspects of this class of CR manifolds in ourstudy. Contents . Introduction and statement of the results . . Introduction and Motivation . . Main theorems . . Applications . . Kawasaki’s Hirzebruch-Riemann-Roch and Grauert-Riemenschneider criterionfor orbifold line bundles . . Examples . . Proof of Theorem 1.2 . . The idea of the proofs of Theorem 1.3, Theorem 1.10 and Corollary 1.13 S ACTION 2
Part I: Preparatory foundations . Preliminaries . . Some standard notations . . Set up and terminology . . Tangential de Rham cohomology group, Tangential Chern character andTangential Todd class . . BRT trivializations and rigid geometric objects . A Hodge theory for (cid:3) ( q ) b,m . Modified Kohn Laplacian (Spin c Kohn Laplacian )5 . Asymptotic expansions for the heat kernels of the modified Kohn Laplacians . . Heat kernels of the modified Kodaira Laplacians on BRT trivializations . . Heat kernels of the modified Kohn Laplacians (Spin c Kohn Laplacians ) Part II: Proofs of main theorems . Proofs of Theorems 1.3 and 1.10 . Trace integrals and proof of Theorem 1.14 . . A setup, including a comparison with recent developments . . Local angular integral . . Global angular integral . . Patching up angular integrals over X; proof for the simple type . . Types for S stratifications; proof for the general typeReferences1. I NTRODUCTION AND STATEMENT OF THE RESULTS
Introduction and Motivation.
Let ( X, T , X ) be a compact (with no boundary) CR manifold ofdimension n + 1 and let ∂ b : Ω ,q ( X ) → Ω ,q +1 ( X ) be the tangential Cauchy-Riemann operator. For asmooth function u , we say u is CR if ∂ b u = 0 . In CR geometry, it is crucial to be able to produce manyCR functions. When X is strongly pseudoconvex and the dimension of X is greater than or equalto five, it is well-known that the space of global smooth CR functions H b ( X ) is infinite dimensional.When X is weakly pseudoconvex or the Levi form of X has negative eigenvalues, the space of globalCR functions could be trivial. In general, it is very difficult to determine when the space H b ( X ) islarge.A clue to the above phenomenon arises from the following. By ∂ b = 0 , one has the ∂ b -complex: · · · → Ω ,q − ( X ) → Ω ,q ( X ) → Ω ,q +1 ( X ) → · · · and the Kohn-Rossi cohomology group: H qb ( X ) := Ker ∂ b :Ω ,q ( X ) → Ω ,q +1 ( X )Im ∂ b :Ω ,q − ( X ) → Ω ,q ( X ) . As in complex geometry, to understand the space H b ( X ) , one could try toestablish, in the CR case, a Hirzebruch-Riemann-Roch theorem for n P j =0 ( − j dim H jb ( X ) , an analogueof the Euler characteristic, and to prove vanishing theorems for H jb ( X ) , j ≥ . The first difficulty with EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 3 such an approach lies in the fact that dim H jb ( X ) could be infinite for some j . Let’s say more about thisin the following.If X is strongly pseudoconvex and of dimension ≥ , it is well-known that ∂ b : Ω , ( X ) → Ω , ( X ) is not hypoelliptic and for q ≥ , ∂ b : Ω ,q ( X ) → Ω ,q +1 ( X ) is hypoelliptic so that dim H b ( X ) = ∞ and dim H qb ( X ) < ∞ for q ≥ . In other cases if the Levi form of X has exactly one negative, n − positive eigenvalues on X and n > , it is well-known that dim H b ( X ) = ∞ , dim H n − b ( X ) = ∞ and dim H qb ( X ) < ∞ , for q / ∈ { , n − } . With these possibly infinite dimensional spaces, we have thetrouble with defining the Euler characteristic n P j =0 ( − j dim H jb ( X ) properly.Another line of thought lies in the fact that the space H qb ( X ) is related to the Kohn Laplacian (cid:3) ( q ) b = ∂ ∗ b ∂ b + ∂ b ∂ ∗ b : Ω ,q ( X ) → Ω ,q ( X ) . One can try to define the associated heat operator e − t (cid:3) ( q ) b by using spectral theory and then the small t behavior of e − t (cid:3) ( q ) b is closely related to the dimensionof H qb ( X ) . Unfortunately without any Levi curvature assumption, (cid:3) ( q ) b is not hypoelliptic in generaland it is unclear how to determine the small t behavior of e − t (cid:3) ( q ) b . Even if (cid:3) ( q ) b is hypoelliptic, it is stilldifficult to calculate the local density.We are led to ask the following question. Question . Can we establish some kind of heat kernel asymptotic expansions for Kohn Laplacian andobtain a CR Hirzebruch-Riemann-Roch theorem (not necessarily the usual ones) on some class of CRmanifolds?
It turns out that the class of CR manifolds with S action is a natural choice for the above ques-tion. On this class of CR manifolds, the geometrical significance of Kohn’s (cid:3) b operator in connectionwith transversally elliptic operators initiated by Atiyah and Singer [1], has been mentioned in theseminal work of Folland and Kohn ([34], p.113). Three dimensional (strongly pseudoconvex) CRmanifolds with S action have been intensively studied back to 1990s in relation to the CR embed-dability problem. We refer the reader to the works [25] and [48] of Charles Epstein and LaszloLempert respectively. (see more comments on this in Section 1.3). Another related paper is about CRstructure on Seifert manifolds by Kamishima and Tsuboi [44] (cf. Remark 1.15). In our present paperthe CR manifold with S action is not restricted to the three dimensional case.To motivate our approach, let’s first look at the case which can be reduced to complex geometry.Consider a compact complex manifold M of dimension n and let ( L, h L ) → M be a holomorpic linebundle over M , where h L denotes a Hermitian fiber metric of L . Let ( L ∗ , h L ∗ ) → M be the dualbundle of ( L, h L ) and put X = n v ∈ L ∗ ; | v | h L ∗ = 1 o . We call X the circle bundle of ( L ∗ , h L ∗ ) . It isclear that X is a compact CR manifold of dimension n + 1 . Clearly X is equipped with a natural(globally free) S action (by acting on the circular fiber).Let T ∈ C ∞ ( X, T X ) be the real vector field induced by the S action, that is, T u = ∂∂θ ( u ( e − iθ ◦ x )) | θ =0 , u ∈ C ∞ ( X ) . This S action is CR and transversal , i.e. [ T, C ∞ ( X, T , X )] ⊂ C ∞ ( X, T , X ) and C T ( x ) ⊕ T , x X ⊕ T , x X = C T x X respectively. For each m ∈ Z and q = 0 , , , . . . , n , put Ω ,qm ( X ) : = (cid:8) u ∈ Ω ,q ( X ); T u = − imu (cid:9) = n u ∈ Ω ,q ( X ); u ( e − iθ ◦ x ) = e − imθ u ( x ) , ∀ θ ∈ [0 , π [ o . Since ∂ b T = T ∂ b , we have ∂ b,m = ∂ b : Ω ,qm ( X ) → Ω ,q +1 m ( X ) . We consider the cohomology group: H qb,m ( X ) := Ker ∂ b,m :Ω ,qm ( X ) → Ω ,q +1 m ( X )Im ∂ b,m :Ω ,q − m ( X ) → Ω ,qm ( X ) , and call it the m -th S Fourier component of the Kohn-Rossicohomology group.The following result can be viewed as the starting point of this paper. Note Ω ,q ( M, L m ) denotes thespace of smooth sections of (0 , q ) forms on M with values in L m ( m -th power of L ) and H q ( M, L m ) the q -th ∂ -Dolbeault cohomology group with values in L m . EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 4
Theorem 1.2.
For every q = 0 , , , . . . , n , and every m ∈ Z , there is a bijective map A ( q ) m : Ω ,qm ( X ) → Ω ,q ( M, L m ) such that A ( q +1) m ∂ b,m = ∂A ( q ) m on Ω ,qm ( X ) . Hence, Ω ,qm ( X ) ∼ = Ω ,q ( M, L m ) and H qb,m ( X ) ∼ = H q ( M, L m ) . In particular dim H qb,m ( X ) < ∞ and n P j =0 ( − j dim H jb,m ( X ) = n P j =0 ( − j dim H j ( M, L m ) . Theorem 1.2 is probably known to the experts. As a precise reference is not easily available (see,however, Folland and Kohn [34] p.113), we will give a proof of Theorem 1.2 in Section 1.6 for theconvenience of the reader.In this paper by
Kodaira Laplacian we mean the Laplacian (cid:3) ( q ) m on L m -valued (0 , q ) forms (on M )associated with the ∂ operator, a term we borrow from the work of Ma and Marinescu [49]. Let e − t (cid:3) ( q ) m be the associated heat operator. It is well-known that e − t (cid:3) ( q ) m admits an asymptotic expansionas t → + . Consider B m ( t ) := ( A ( q ) m ) − ◦ e − t (cid:3) ( q ) m ◦ A ( q ) m ( A ( q ) m as in the theorem above). Let (cid:3) ( q ) b,m be theKohn Laplacian (on X ) acting on (the m -th S Fourier component of) (0 , q ) forms, with e − t (cid:3) ( q ) b,m theassociated heat operator.A word of caution is in order. We made no use of metrics for stating Theorem 1.2. However, todefine those Laplacians above an appropriate choice of metrics is needed (for adjoint of an operator)so that A ( q ) m of Theorem 1.2 also preserves the chosen metrics. With this set up it is fundamental that(cf. Proposition 5.1)(1.1) e − t (cid:3) ( q ) b,m = (( A ( q ) m ) − ◦ e − t (cid:3) ( q ) m ◦ A ( q ) m ) ◦ Q m = B m ( t ) ◦ Q m = Q m ◦ B m ( t ) ◦ Q m , where Q m : Ω ,q ( X ) → Ω ,qm ( X ) is the orthogonal projection. Hence the asymptotic expansion of e − t (cid:3) ( q ) m and (1.1) lead to an asymptotic expansion(1.2) e − t (cid:3) ( q ) b,m ( x, x ) ∼ t − n a ( q ) n ( x ) + t − n +1 a ( q ) n − ( x ) + · · · . One goal of this work is to establish a formula similar to (1.2) (which is however not exactly of thisform) on any CR manifold with S action. More precisely, due to the assumption that the S actionis only locally free, it turns out that e − t (cid:3) ( q ) b,m ( x, x ) cannot have the standard asymptotic expansion as(1.2). Rather, our asymptotic expansion involves an unprecedented contribution in terms of a distancefunction from lower dimensional strata of the S action. (See (1.18) in Theorem 1.3 for details andfor our first main result.) It should be emphasized that no pseudoconvexity condition is assumed.Roughly speaking, on the regular part of X we have(1.3) e − t (cid:3) ( q ) b,m ( x, x ) ∼ t − n a ( q ) n ( x ) + t − n +1 a ( q ) n − ( x ) + · · · mod O (cid:16) t − n e − ε d ( x,X sing )2 t (cid:17) . On the whole X we have, however,(1.4) e − t (cid:3) ( q ) b,m ( x, x ) ∼ t − n A ( q ) n ( t, x ) + t − n +1 A ( q ) n − ( t, x ) + · · · . The difference between (1.4) and (1.3) lies in that A ( q ) s ( t, x ) in (1.4) cannot be t -independent forall s and are not canonically determined (by our method) while a ( q ) s ( x ) in (1.3) are t -independentfor all s and are canonically determined. This ( t -dependence nature so introduced) presents a greatdistinction between our asymptotic expansion and those in the previous literature, and this distinctpoint of departure appears to have a big influence on the formulation and proof of the relevant indextheorems and trace integrals. See Section 7 for more comments.In addition to the introduction of a distance function ˆ d in (1.3) our generalization has anotherfeature, which is pertinent to the third topic of this paper, as follows. A heat kernel result for orbifoldsobtained in 2008 by Dryden, Gordon, Greenwald and Webb for the case of Laplacian on functions(see (1.30) and [20]) and independently by Richardson ([57]) seems to suggest that integrating (1.3)over X is basically a power series in t . See (1.30) for more. To see such a possible connection, one EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 5 considers X as a fiber space over X/S which is then an orbifold, and presumes boldly an analogywith “(1.2) for the orbifold case”. Then by the above result [20], integrating (1.3) over X might givean asymptotic expansion which is a power series at most in the fractional power t of t (cf. Theorem1.14) (while for the case where the S action is globally free, such as in the circle bundle above, theasymptotic expansion is expressed in the integral power of t ). However, our further study shows thatthe coefficients of t j for j being half-integral necessarily vanish in our present case (irrespective ofthe local or global freeness of the S action). Despite that there is no nontrivial fractional power inthe t -expansion, the corrections /contributions associated with the stratification of the locally free S action do arise nontrivially in a proper sense. Some explicit computations about these extra terms areworked out in the main result of the final section (Section 7) regarded as the third topic of this paper.As far as the asymptotic expansion is concerned, we remark that the approach of using KodairaLaplacian on M (downstairs) as done above is no longer applicable to the general CR case, as thecontribution of a distance function on X involved in our expansion cannot be easily forseen by useof objects in the space downstairs (an orbifold in general). (However, for trace integrals on invariantfunctions, cf. Section 7, like P m e − tλ m denoted by I ( t ) in certain Riemannian cases, I ( t ) has beenstudied asymptotically with the help of the underlying/quotient manifold/orbifold, cf. [57, p. 2316-2317]. See also Proposition 5.1, Remark 5.3.) We must work on the entire X from scratch with theoperator being only transversally elliptic (on X ). (See HRR theorem below for another instance ofthis idea.) Furthermore, as we make no assumption on (strong) pseudoconvexity of X , this rendersthe techniques usually useful in this direction by previous work (e.g. [3]) hardly adequate in our case.Our current approach is essentially independent of the previous methods. This technicality partlyaccounts for the length of the present paper (see Section 1.7 for an outline of proof and Section 7 fora comparison with the previous work).We expect that the coefficients a ( q ) s ( x ) in (1.3) are related to some geometric quantities. For q =0 , function case with strong pseudoconvexity, we refer the reader to the paper of Beals, Greiner, andStanton [3]. In this regard, Chern-Moser invariants (see [17], and [5] for a related question posed inthe end of the paper) or Tanaka-Webster invariants (see [59] or [61]) should be used to express thesecoefficients. In our present situation (without assumptions on pseudoconvexity) it is however morenatural to use geometric quantities adapted to the S invariance property, so that a notion of tangentialcurvature arises (with the associated tangential characteristic forms, cf. Section 2.3) and enters intothe coefficients of our asymptotic expansion. It essentially comes back to the Tanaka-Webster curvaturein the strongly pseudoconvex case (cf. Remark 1.9).The mathematics (existence, asymptotics etc.) of equivariant/transversal heat kernels in the Rie-mannian situation (including that of Riemannian foliations) have been studied in recent years andlast decades. For a comparison between these developments and our results, we postpone the survey,together with that of trace integrals, until Section 7.Back to the special case of the circle bundle X over a compact complex manifold M , the Hirzebruch-Riemann-Roch Theorem or Atiyah-Singer index Theorem, together with Theorem 1.2, tells us that(1.5) n X j =0 ( − j dim H jb,m ( X ) = n X j =0 ( − j dim H j ( M, L m ) = Z M Td ( T , M )ch ( L m ) , in terms of standard characteristic classes on M . Let’s reformulate (1.5) in geometric terms on X rather than on M :(1.6) n X j =0 ( − j dim H jb,m ( X ) = 12 π Z X Td b ( T , X ) ∧ e − m dω π ∧ ω where Td b ( T , X ) denotes the tangential Todd class of T , X and e − m dω π denotes the Chern polyno-mial of the Levi curvature and ω is the uniquely determined global real -form (see Section 2.2 andSection 2.3 for the precise definitions). EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 6
Our second main result turns to any (abstract) CR manifold X with (locally free) S action (butwith no assumption on pseudoconvexity); we see that the above Euler characteristic has an indexinterpretation related to ∂ b + ∂ ∗ b on X (see (3.12) and (3.13)). We are able to establish (1.6) (cf.Corollary 1.13) on such X based on our asymptotic expansion for the heat kernel e − t (cid:3) ( q ) b,m ( x, x ) and atype of McKean-Singer formula on X (see Corollaries 4.8 and 5.15).As an application to complex (orbifold) geometry, it is worth noting a comparison between thepresent result and a result of Kawasaki on Hirzebruch-Riemann-Roch theorem (HRR theorem forshort) on a complex orbifold N ([46]) (which plays the role as our M above). Our formula simpli-fies in the sense that in the original formula of Kawasaki, his part involving the dependence on the(lower dimensional) strata of the orbifold M entirely drops out here, at least for the class of orbifoldsand orbifold line bundles that fit into our assumption (see Theorem 1.27, remarks following it andSubsection 1.5.2 for examples). In our view this simplification does not appear obvious at all withinthe original approach of Kawasaki because by his approach the contributions from the (lower dimen-sional) strata of the orbifold cannot be avoided (unless it is proved to be vanishing) even if the totalspace of the (orbifold) circle bundle is smooth. Conceptually speaking one may attribute such a sim-plification to one’s working on the entire (smooth) X rather than on the downstairs M (as Kawasaki),a strategy already employed for the asymptotic expansion above and proving useful again in this con-text of (CR) index theorem. We remark that the vanishing of the contribution of strata also occurs ina related context studied by these works [55], [33] (see also discussions after Theorem 1.27).In short our second main result (Theorem 1.10) computes a local index density in terms of tangential characteristic forms, which is to show that certain non-trivial contributions (cf. t − n e − ε d ( x,X sing )2 t of(1.3)) in the heat kernel expansion will eventually cancel out in the index density computation. Wecan do this by applying Getzler’s rescaling technique to the off-diagonal estimate (not needed in theclassical index theorems). As, to the best of our knowledge, an appropriate term for such a resultabout the local density hasn’t appeared in the literature yet, we shall follow the classical cases andcall it a local index theorem on these CR manifolds (Corollary 1.13), including Sasakian manifolds ofinterest in String Theory.With reference to the questions in the beginning of this Introduction, for further application of ourresults to CR geometry it is important to produce many CR functions or CR sections. Namely we hopeto know when H b ( X, E ) or H b,m ( X, E ) is large (see Questions 1.17, 1.18 and 1.22 in Section 1.3).Progress towards this circle of questions seems limited (Section 1.3). We can now develop a tool fortackling some of these questions. The idea here is to combine our version of CR index theorem witha sort of vanishing theorem for higher cohomology groups, which is intimately related to a version ofGrauert-Riemenschneider criterion adapted to the CR case. This methodology turns out to be effectivefor those CR manifolds studied in this paper.In Section 1.3 we apply our CR index theorem to prove a CR version of Grauert-Riemenschneidercriterion, and produce many CR functions on a weakly pseudoconvex CR manifold with transversal S action and many CR sections on some class of CR manifolds, which give answers to some long-standingquestions in several complex variables and CR geometry. In Section 1.5 we provide an abundance ofexamples of those CR manifolds studied in the present paper, some of which arise from Brieskornmanifolds (generalized Hopf manifolds).There is another index theory of geometric significance, developed by Charles Epstein. He studiedthe so called relative index of a pair of embeddable CR structures through their Szeg¨o projectors in aseries of papers (see [26], [27], [28], [29] and [30]). On the other hand, Erik van Erp derived an indexformula for subelliptic operators on a contact manifold (see [31], [32]). Moreover, recent work ofParadan and Vergne [55] gave an expression for the index of transversally elliptic operators which is anintegral of compactly supported equivariant form on the cotangent bundle; see also Fitzpatrick [33] for EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 7 related directions. Br¨uning, Kamber and Richardson [12], [13] computed the index of an equivarianttransversally elliptic operator as a sum of integrals over blow ups of the strata of the group action.Finally it is natural to ask for a generalization from the action of S to that of other Lie groups oreven to foliations (cf. Subsection 7.1 for references). As we will discuss in Section 7, the asymptoticexpansion in the form (1.4) as indicated there a sort of remedy for (1.3) by involving a “distancefunction” ˆ d seems to be best illustrated in the S case. It appears also conceivable that these featuresshall be preserved for generalization in a certain (as yet unknown) way. This paper may be presentedor read in token of a prototype for further study into much more complicated, diversified situations.1.2. Main theorems.
We shall now formulate the main results. We refer to Section 2.2 and Sec-tion 2.3 for some notations and terminologies used here. After the background material, we willdiscuss in the sequel i) asymptotic expansions, ii) a local index theorem and iii) trace integrals.1.2.1.
Background.
Let ( X, T , X ) be a compact connected CR manifold with a transversal CR locallyfree S action e − iθ , where T , X is a CR structure of X . X is of dimension n + 1 throughout thispaper.Let T ∈ C ∞ ( X, T X ) be the real vector field induced by the S action and let ω ∈ C ∞ ( X, T ∗ X ) bethe global real one form determined by h ω , T i = 1 , h ω , u i = 0 , for every u ∈ T , X ⊕ T , X .Associated with the S action of X it is natural to consider various geometric objects admitting an S action. In the following, to streamline the exposition we shall freely use the notion of rigid objects:“rigid bundles”, “rigid metrics” etc., and refer to Definitions 2.3, 2.4 and 2.5 for the precise meanings.(See also the work of Baouendi-Rothschild-Treves [4, Definition II.2] for a similar use of this term.) Itsuffices to say here that this notion of rigid objects is nothing but an equivalent way (by using metric)to consider objects (originally defined without assumption on metric) which admit (compatible) S actions (or S invariance, subject to the proper context) provided one starts with a CR manifold withan S action (cf. Theorem 2.11).Henceforth let E be a rigid CR vector bundle over X , equipped with a rigid Hermitian metric h · | · i E . We note that T , X is known to be a rigid complex vector bundle (see the work of Baouendi-Rothschild-Treves [4]) with a rigid Hermitian metric h · | · i satisfying extra properties (not specifiedhere, cf. (2.5)). Let h · | · i E be the Hermitian metric on T ∗ , • X ⊗ E induced by those on E and C T X .Denoting by dv X = dv X ( x ) the volume form on X induced by the Hermitian metric h · | · i on C T X weget the natural global L inner product ( · | · ) E on Ω , • ( X, E ) .As remarked in Introduction, for u ∈ Ω , • ( X, E ) , T u ∈ Ω , • ( X, E ) is defined and T ∂ b = ∂ b T . For m ∈ Z , put Ω , • m ( X, E ) : = (cid:8) u ∈ Ω , • ( X, E ); T u = − imu (cid:9) = n u ∈ Ω , • ( X, E ); ( e − iθ ) ∗ u = e − imθ u, ∀ θ ∈ [0 , π [ o , where ( e − iθ ) ∗ denotes the pull-back by the map e − iθ : X → X of S action.Write L ( X, T ∗ , • X ⊗ E ) (resp. L m ( X, T ∗ , • X ⊗ E ) ) for the L -completion of Ω , • ( X, E ) (resp. Ω , • m ( X, E ) ) with respect to ( · | · ) E .By T ∂ b = ∂ b T one defines ∂ b,m : Ω , • m ( X, E ) → Ω , • m ( X, E ) as the restriction of ∂ b on Ω , • m . Write ∂ ∗ b : Ω , • ( X, E ) → Ω , • ( X, E ) , resp. ∂ ∗ b,m : Ω , • m ( X, E ) → Ω , • m ( X, E ) for the formal adjoint of ∂ b (under ( · | · ) E ), resp. ∂ b,m . Since h · | · i E and h · | · i are rigid, one sees T ∂ ∗ b = ∂ ∗ b T on Ω , • ( X, E ) ,∂ ∗ b,m = ∂ ∗ b | Ω , • m : Ω , • m ( X, E ) → Ω , • m ( X, E ) , ∀ m ∈ Z . (1.7) EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 8
Let A m : Ω , • m ( X, E ) → Ω , • m ( X, E ) be a certain smooth zeroth order operator with T A m = A m T and A m : Ω , ± m ( X, E ) → Ω , ∓ m ( X, E ) (arising from a CR version of Spin c Dirac operator, cf. Definition 4.3).Put(1.8) e D b,m := ∂ b,m + ∂ ∗ b,m + A m : Ω , • ( X, E ) → Ω , • ( X, E ) and let(1.9) e D ∗ b,m : Ω , • ( X, E ) → Ω , • ( X, E ) be the formal adjoint of e D b,m (with respect to ( · | · ) E ).We have e (cid:3) b,m , given by(1.10) e (cid:3) b,m := e D ∗ b,m e D b,m : Ω , • ( X, E ) → Ω , • ( X, E ) which denotes the m -th modified Kohn Laplacian, thought of as
Spin c Kohn Laplacian (cf. Defini-tion 4.3 and the paragraph below it). We extend e (cid:3) b,m by(1.11) e (cid:3) b,m : Dom e (cid:3) b,m ( ⊂ L m ( X, T ∗ , • X ⊗ E )) → L m ( X, T ∗ , • X ⊗ E ) , with Dom e (cid:3) b,m := { u ∈ L m ( X, T ∗ , • X ⊗ E ); e (cid:3) b,m u ∈ L m ( X, T ∗ , • X ⊗ E ) } in which e (cid:3) b,m u is definedin the sense of distribution.We will show in Section 3 that e (cid:3) b,m is self-adjoint, Spec e (cid:3) b,m is a discrete subset of [0 , ∞ [ andfor ν ∈ Spec e (cid:3) b,m , ν is an eigenvalue of e (cid:3) b,m with finite multiplicities d ν < ∞ . Let (cid:8) f ν , . . . , f νd ν (cid:9) be an orthonormal frame for the eigenspace of e (cid:3) b,m with eigenvalue ν . The (smooth) heat kernel e − t e (cid:3) b,m ( x, y ) can be given by(1.12) e − t e (cid:3) b,m ( x, y ) = X ν ∈ Spec e (cid:3) b,m d ν X j =1 e − νt f νj ( x ) ∧ ( f νj ( y )) † , where f νj ( x ) ∧ ( f νj ( y )) † denotes the linear map: f νj ( x ) ∧ ( f νj ( y )) † : T ∗ , • y X ⊗ E y → T ∗ , • x X ⊗ E x ,u ( y ) ∈ T ∗ , • y X ⊗ E y → f νj ( x ) h u ( y ) | f νj ( y ) i E ∈ T ∗ , • x X ⊗ E x . Let e − t e (cid:3) b,m : L ( X, T ∗ , • X ⊗ E ) → L m ( X, T ∗ , • X ⊗ E ) be the (continuous) operator associated withthe distribution kernel e − t e (cid:3) b,m ( x, y ) .Let e ( x ) , . . . , e d ( x ) be an orthonormal frame of T ∗ ,qx X ⊗ E x ( q = 0 , , . . . , n ), and A ∈ End ( T ∗ , • x X ⊗ E x ) . Put Tr (q) A := P dj =1 h Ae j | e j i E and set Tr A := n X j =0 Tr (j) A, STr A := n X j =0 ( − j Tr (j) A. (1.13)Let ∇ T X be the Levi-Civita connection on
T X (with respect to h · | · i ). Then T , X is equipped witha connection ∇ T , X := P T , X ∇ T X where P T , X be the projection from C T X onto T , X .Let ∇ E be the connection on E induced by h · | · i E (see Theorem 2.12). Let Td b ( ∇ T , X , T , X ) denote the representative of the tangential Todd class of T , X and ch b ( ∇ E , E ) the representative ofthe tangential Chern character of E (see Section 2.3 for tangential classes).In what follows we aim to define a distance function ˆ d which plays an important role (for theasymptotic expansion) in this paper. For x ∈ X , we say that the period of x is πℓ , ℓ ∈ N provided that EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 9 e − iθ ◦ x = x for every < θ < πℓ and e − i πℓ ◦ x = x . Put, for each ℓ ∈ N ,(1.14) X ℓ = (cid:8) x ∈ X ; the period of x is πℓ (cid:9) and let p = min { ℓ ∈ N ; X ℓ = ∅} . We call X p = X p the principal stratum . It is well-known that if X is connected, then X p is an open and dense subset of X (see Proposition 1.24 in Meinrenken [51]and Duistermaat-Heckman [24]). Assume X = X p S X p S · · · S X p k , p =: p < p < · · · < p k . Put X sing = X := S kj =2 X p j , X r sing := S kj = r +1 X p j , k − ≥ r ≥ . Set X k sing := ∅ . Note p | p j for ≤ j ≤ k (cf. Remark 1.16).Let d ( · , · ) denotes the standard Riemannian distance with respect to the given Hermitian metric.Take ζ (1.15) < ζ < inf (cid:26) πp k , (cid:12)(cid:12)(cid:12)(cid:12) πp r − πp r +1 (cid:12)(cid:12)(cid:12)(cid:12) , r = 1 , . . . , k − (cid:27) . Set, for x ∈ X and r = 1 , , . . . , k ,(1.16) ˆ d ζ ( x, X r sing ) := inf (cid:26) d ( x, e − iθ x ); ζ ≤ θ ≤ πp r − ζ (cid:27) . This notation reflects the fact that ˆ d ζ ( x, X r sing ) is equivalent to the ordinary distance d ( x, X r sing ) (seebelow). Note by definition ˆ d ζ ( x, X k sing ) (= ˆ d ζ ( x, ∅ )) > for all x ∈ X . We remark that for any < ζ, ζ satisfying (1.15), ˆ d ζ ( x, X r sing ) and ˆ d ζ ( x, X r sing ) are equivalent (as far as the estimate in Theorem 1.3below is concerned). We shall denote ˆ d ( x, X r sing ) := ˆ d ζ ( x, X r sing ) .Remark that, by examining the definition ˆ d ( x, X r sing ) = 0 if and only if x ∈ X r sing . Further, for ε > there is a δ > such that ˆ d ( x, X r sing ) ≥ δ provided x ∈ X satisfies (the ordinary distance) d ( x, X r sing ) ≥ ε . It is thus convenient to think of ˆ d ( x, X r sing ) as a distance function from x to X r sing .Indeed in Theorem 6.7 for a strongly pseudoconvex X there is a constant C ≥ such that C d ( x, X r sing ) ≤ ˆ d ( x, X r sing ) ≤ Cd ( x, X r sing ) , ∀ x ∈ X. Asymptotic expansion of the heat kernel e − t e (cid:3) b,m ( x, x ) . With the distance function ˆ d , we state thefirst main result of this paper (see Section 6 for a proof). Theorem 1.3.
Suppose ( X, T , X ) is a compact, connected CR manifold (of dimension n + 1 ) witha transversal CR locally free S action. With the notations above, there exist a s ( t, x ) (= a s,m ( t, x )) ∈ C ∞ ( R + × X, End ( T ∗ , • X ⊗ E )) with | a s ( t, x ) | ≤ C on R + × X where C > is independent of t , s = n, n − , . . . , such that (1.17) e − t e (cid:3) b,m ( x, x ) ∼ t − n a n ( t, x ) + t − n +1 a n − ( t, x ) + · · · as t → + . (See Definition 5.4 for “ ∼ ”.)Moreover, there exist α s ( x ) (= α s,m ( x )) ∈ C ∞ ( X, End ( T ∗ , • X ⊗ E )) , s = n, n − , . . . , satisfying thefollowing property. Given any differential operator P ℓ : C ∞ ( X, T ∗ , • X ⊗ E ) → C ∞ ( X, T ∗ , • X ⊗ E ) oforder ℓ ∈ N , there exist ε > and C > such that (1.18) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ℓ (cid:16) a s ( t, x ) − (cid:0) p r X s =1 e π ( s − pr mi (cid:1) α s ( x ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C t − ℓ e − ε d ( x,Xr sing )2 t , ∀ t ∈ R + , ∀ x ∈ X p r r = 1 , . . . , k . The following is immediate from the proof of Theorem 1.3.
EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 10
Corollary 1.4.
Suppose ( X, T , X ) is a compact, connected CR manifold with a transversal CR locally free S action. With the notations above, for any r = 1 , . . . , k , any differential operator P ℓ : C ∞ ( X, T ∗ , • X ⊗ E ) → C ∞ ( X, T ∗ , • X ⊗ E ) of order ℓ ∈ N , every N ∈ N with N ≥ N ( n ) for some N ( n ) , there are ε > , δ > and C N > such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ℓ (cid:16) e − t e (cid:3) b,m ( x, x ) − (cid:0) p r X s =1 e π ( s − pr mi (cid:1) N X j =0 t − n + j α n − j ( x ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C N (cid:16) t − n + N +1 − ℓ + t − n − ℓ e − ε d ( x,Xr sing )2 t (cid:17) , ∀ x ∈ X p r , ∀ < t < δ. (1.19)In the following we supplement these results with a number of remarks before going further. Remark . ( p r P s =1 e π ( s − pr mi ) = p r if p r | m ; ( p r P s =1 e π ( s − pr mi ) = 0 if p r m . Remark . We shall now see that if one wants an asymptotic expansion of e − t e (cid:3) b,m ( x, x ) to be validaround each x ∈ X (cf. Definition 5.4), then (1.17) is basically optimal (i.e. in general, a s ( t, x ) cannotbe t -independent for all s ). For U open with U ⊂ X p , a tradition-like formula (assuming p = 1 forsimplicity)(1.20) e − t e (cid:3) b,m ( x, x ) ∼ C (cid:0) t − n α n ( x ) + t − n +1 α n − ( x ) + · · · (cid:1) is valid for x ∈ U and C = 1 (as follows from (1.18) for l = 0 ) whereas for x ∈ X p r , r ≥ , anasymptotic expansion (for p r | m ) with C = p r is valid around an open subset ( ∋ x ) of the stratum X p r .Since e − t e (cid:3) b,m ( x, y ) is going to be a well defined smooth kernel, it is easily seen that those functions α s ( x ) ( s = n, n − , · · · ) satisfying Theorem 1.3 are unique (if they exist). (We notice that a s ( t, x ) in(1.17) are not canonically defined by our method which is subject to choice of BRT trivializations, cf.(5.40) and Subsection 2.4.) In short, the above suggests that an asymptotic expansion of the form as(1.20) can only be true in the piecewise sense with respect to strata. See also Subsection 7.1.To confirm this, one uses Theorem 1.14 (see Theorems 7.20 and 7.24 for a more precise version) bynoting R X Tr α + s ( x ) dv X ( x ) = S +1 ,s in Theorem 7.20 (which is taking the “even” part of the Laplacian).Hence one can interpret the trace integral result (obtained by integrating Tr e − t e (cid:3) b,m ( x, x ) over X ) asone that gives extra nonzero correction terms , cf. the second line in (1.32) or the third line in (7.50).It follows that if there exists a global asymptotic expansion (not just in the piecewise sense) such as(1.17), then not all of a s ( t, x ) can be independent of t . Otherwise, if all a s ( t, x ) are independent of t ,it would be of the form (1.20) globally by assumption ( C = 1 if p = 1 ), so by integrating the traceover X , there would be no correction terms as discussed above. To say more, e − t e (cid:3) b,m ( x, x ) cannothave any asymptotic expansion of the form t m β m ( x ) + t m β m ( x ) + · · · (globally) m < m <, · · · ∈ R , β m ( x ) , β m ( x ) , · · · continuous functions on X . Otherwise by equating it to (1.17), each a s ( t, x ) would be rendered independent of t , absurd as just remarked (see the next remark for argumentindependent of Theorems 1.14, 7.20).The next remark shows that a s ( t, x ) for the particular s = n must be dependent on t (nontrivially).This part will not use Theorem 1.14. Remark . In the above remark a certain discontinuity in the form (1.20) for, say x ∈ X p and x ∈ X p seems to appear. We shall now explore it. If the (Gaussian-like) term to the right of (1.18) isexamined, it arises from a precise integral (see (6.8)). To show that this integral is generally nontrivial,regardless of whether our estimate given by (1.18) is a fine or crude one, we are actually going toshow that the term for s = n in (1.18) a n ( t, x ) − p r X s =1 e π ( s − pr mi α n ( x ) EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 11 is nontrivial. For the sake of illustration we assume that X = X S X , that is p = 1 and p = 2 , andtake m to be an even number. For x ∈ X , by (1.18) (for l =0) and p = 1 , we see that a n ( t, x ) = α n ( x ) + r n ( t, x ) (1.21) | r n ( t, x ) | . e − ε d ( x,X sing )2 t . As our α n ( x ) essentially arises from a local Kodaira Laplacian (see (6.1), similar to discussion afterTheorem 1.2), it is well known that α n ( x ) , as the coefficient of the leading term (in the t -expansionof the heat kernel for Kodaira Laplacian), is constant in x with Tr α n > (cf. [37, Lemma 4.1.4 andSection 4.4]). By continuity ( a s and α s being globally continuous functions)(1.22) a n ( t, x ) = α n ( x ) + r n ( t, x ) remains true on X . For x ∈ X , the estimate of (1.18) is given by (with p = 2 and discussion after(1.16) for ˆ d ( x , ∅ ) > )(1.23) a n ( t, x ) = 2 α n ( x ) + O ( t ∞ ) . By (1.22) and (1.23) it follows r n ( t, x ) = α n ( x ) + O ( t ∞ ) so r n ( t, x ) ≈ α n ( x ) around x as t → ,giving | r n ( t, x ) | ≥ ǫ > nearby x for some constant ǫ independent of x and t . But this would beabsurd by (1.21) if r n were independent of t (taking x ( ∈ X , = x ) near x so that | r n ( t, x ) | ≥ ǫ andletting t → in (1.21)). Hence a n ( t, x ) cannot be independent of t either, as desired. Remark . To discuss the estimate (1.19), let’s take ˆ d in (1.19) to be d for convenience (as remarkedpreviously ˆ d is equivalent to the ordinary distance function d at least in the strongly pseudoconvexcase, cf. Theorem 6.7). Take P l = id (so l = 0 ). The term to the rightmost of (1.19) appears as aGaussianlike term. As t → , this term tends to a sort of Dirac delta function supported along thestrata X r sing (with an extra singular factor t − a − , a = dim X r sing ). This may conceptually explain thepiecewise continuity nature just discussed in Remarks 1.6 and 1.7 if the asymptotic expansion is tobe expressed in something, without t -dependence, such as α s ( x ) . Conversely, the estimate as (1.19)involving a type of Dirac delta function is conceptually reasonable under the piecewise continuityphenomenon in terms of α s ( x ) . For more about this, some quantitative information may be availableby Theorems 1.14, 7.20 and 7.24. Remark . We make a short comment on the coefficients a s ( t, x ) or α j ( x ) in (1.18) (the differencebetween a s ( t, x ) and p α s ( x ) (at a given x ∈ X p ) is O ( t ∞ ) by (1.18); this is partly explained concep-tually right below). For the standard (elliptic) case (of Dirac type) it is well-known that the coefficientsof a heat kernel along the diagonal (by taking trace) are expressible in terms of the curvature and itscovariant derivatives (e.g. [37]). In our transversally elliptic case (without bundle E for simplicity)if S action is globally free, it follows from the standard case above (cf. (1.1)-(1.2)) that these co-efficients of the (transversal) heat kernel are expressible in terms of the tangential curvature (and itsderivatives) (cf. Section 2.3). In the locally free case the same results can be achieved in view of theproof of Theorem 1.3, which basically arises from a procedure of patching and successive approxima-tions based on the local (transversal) heat kernels that give the asymptotic approximations of the final(transversal) heat kernel (see Section 1.7 for details of an outline). Since the local kernels can be soexpressed as just said (at least on the principal stratum), it follows from the asymptotic approxima-tion (e.g. Theorems 2.23 and 2.30 of [6] or Theorem 5.14 in our case) that the same (expression intangential curvature and its derivatives) can be said for the global kernel (on the principal stratumthen followed by continuous extension of this global kernel on X ). It is also of interest to consider theintegral version of these coefficients, which is the topic of Section 7 of this paper.1.2.3. A local index theorem for CR manifolds with S action. Here we discuss issues related to theindex theorem we will prove. We recall that the term to the left of the inequality in (1.18) is basicallynontrivial by Remark 1.7. In our formulation of index theorems, the contribution arising from such a
EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 12 term is expected to be removed. This can be done when e (cid:3) b,m is the Spin c Kohn Laplacian (cf. (4.12)).In this case, we show that taking supertrace in (1.19) ( P l = id ) and applying Getzler’s rescalingtechnique to the off-diagonal estimate (see Subsection 1.7.3 for more) yield that the singular part t − n to the rightmost of (1.19) can be removed (see Subsection 1.7.4 and Section 6 for a proof). Moreprecisely Theorem 1.10.
Suppose ( X, T , X ) is a compact, connected CR manifold with a transversal CR locallyfree S action. With the notations above, if e (cid:3) b,m is the Spin c Kohn Laplacian (see (4.12) ), then for r = 1 , . . . , k and every N ∈ N with N ≥ N ( n ) for some N ( n ) , there are ε > , δ > and C N > such that ( STr denoting supertrace, cf. (1.13) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) STr e − t e (cid:3) b,m ( x, x ) − (cid:0) p r X s =1 e π ( s − pr mi (cid:1) N X j =0 t − n + j STr α n − j ( x ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C N (cid:16) t − n + N +1 + e − ε d ( x,Xr sing )2 t (cid:17) , ∀ < t < δ, ∀ x ∈ X p r , (1.24) and n X ℓ =0 t − ℓ STr α ℓ ( x ) dv X ( x )= 12 π h Td b ( ∇ T , X , T , X ) ∧ ch b ( ∇ E , E ) ∧ e − m dω π ∧ ω i n +1 ( x ) (1.25) where [ ... ] | n +1 to the right denotes the part of (2 n + 1) -form. As Spin c objects can be simplified in the K¨ahler case, so can the Spin c Kohn Laplacian in the
CRK¨ahler case , to which we turn now.
Definition 1.11.
We say that X is CR K¨ahler if there is a closed form Θ ∈ C ∞ ( X, T ∗ , X ) such that Θ( Z, Z ) > , for all Z ∈ C ∞ ( X, T , X ) . We call Θ a CR K¨ahler form on X .When X is a strongly pseudoconvex CR manifold with a transversal CR locally free S action, theclosed form dω satisfies dω ( Z, Z ) > , for all Z ∈ C ∞ ( X, T , X ) . Hence X is CR K¨ahler.A quasi-regular Sasakian manifold is also a CR K¨ahler manifold. We recall that for a compact smoothmanifold X of dim X = 2 n + 1 , n ≥ , the triple ( X, g, α ) where g is a Riemannian metric and α is areal 1-form is called a Sasakian manifold if the cone C ( X ) = { ( x, t ) ∈ X × R > } is a K¨ahler manifoldwith complex struture J and K¨ahler form t dα + 2 tdt ∧ α compatible with the metric t g + dt ⊗ dt (see[7], [9], [53]). As a consequence, X is a compact strongly pseudoconvex CR manifold and the Reebvector field ξ , defined by α ( · ) = g ( ξ, · ) , induces a transversal CR R action on X . If the orbits of this R action are compact, the Sasakian structure is called quasi-regular. In this case, the Reeb vector fieldgenerates a locally free transversal CR S action on X . We can thus identify a compact quasi-regularSasakian manifold with a compact strongly pseudoconvex CR manifold ( X, T , X ) equipped with atransversal CR locally free S action such that the induced vector field of the S action coincides withthe Reeb vector field on X (see [52], [53]).Let X be a CR K¨ahler manifold with a transversal CR locally free S action. If h · | · i is induced by aCR K¨ahler form on X , then (cid:3) b,m is equal to the Spin c Kohn Laplacian. By Theorem 1.10, we immedi-ately obtain a version of local index theorem on CR K¨ahler manifolds with transversal CR locally free S action (which include the compact quasi-regular Sasakian manifolds as a special case by above).These results are discussed below.For a proof of the following, see the beginning of Subsection 1.7.4 and the discussion leading toProposition 5.8): Corollary 1.12. (CR K¨ahler case of Theorem 1.10) Suppose ( X, T , X ) is a compact, connected CRK¨ahler manifold with a transversal CR locally free S action and assume that h · | · i is induced by a CR EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 13
K¨ahler form on X . With the notations above, for r = 1 , . . . , k and every N ∈ N with N ≥ N ( n ) forsome N ( n ) , there are ε > , δ > and C N > such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) STr e − t (cid:3) b,m ( x, x ) − p r X s =1 e π ( s − pr mi N X j =0 t − n + j STr α n − j ( x ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C N (cid:16) t − n + N +1 + e − ε d ( x,Xr sing )2 t (cid:17) , ∀ < t < δ, ∀ x ∈ X p r , (1.26) and n X ℓ =0 t − ℓ STr α ℓ ( x ) dv X ( x )= 12 π h Td b ( ∇ T , X , T , X ) ∧ ch b ( ∇ E , E ) ∧ e − m dω π ∧ ω i n +1 ( x ) . (1.27)We are in a position to state an index theorem (including a local index theorem in the CR K¨ahlercase). Recall ∂ b,m := ∂ b : Ω ,qm ( X, E ) → Ω ,q +1 m ( X, E ) , m ∈ Z , and a ∂ b,m -complex: ∂ b,m : · · · → Ω ,q − m ( X, E ) → Ω ,qm ( X, E ) → Ω ,q +1 m ( X, E ) → · · · . The q -th ∂ b,m Kohn-Rossi cohomology group (regarded as the m -th Fourier compoment of the ordinary q -th Kohn-Rossi cohomology group) is H qb,m ( X, E ) := Ker ∂ b,m : Ω ,qm ( X, E ) → Ω ,q +1 m ( X, E )Im ∂ b,m : Ω ,q − m ( X, E ) → Ω ,qm ( X, E ) . We will prove in Theorem 3.7 that there holds dim H qb,m ( X, E ) < ∞ (for each m ∈ Z and q =0 , , , . . . , n ) without any Levi curvature assumption.In Corollary 4.8 (see also Remark 4.9) we have a McKean-Singer type formula in our CR case: forevery t > ,(1.28) n X j =0 ( − j dim H jb,m ( X, E ) = Z X STr e − t e (cid:3) b,m ( x, x ) dv X . Combining (1.28), (1.24) and (1.25) and noting e − ε d ( x,X sing )2 t is bounded by and rapidly decaysto for x in the principal stratum as t → , we conclude the following form of an index theorem onour CR manifolds (see Section 2.3 for the precise meanings of Td b ( T , X ) and ch b ( E ) below): Corollary 1.13. (CR Index Theorem, cf. Corollary 6.5) Suppose ( X, T , X ) is a compact, connected CRmanifold with a transversal CR locally free S action. Then n X j =0 ( − j dim H jb,m ( X, E )= ( p (= p ) X s =1 e π ( s − p mi ) 12 π Z X Td b ( T , X ) ∧ ch b ( E ) ∧ e − m dω π ∧ ω , (1.29) where Td b ( T , X ) denotes the tangential Todd class of T , X and ch b ( E ) denotes the tangential Cherncharacter of E . For a connection with other works on index theorems by different formulations and methods, werefer to comments that come after Theorem 1.27 and to the sixth paragraph in Subsection 7.1.
EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 14
Trace integrals in terms of geometry of the S stratification. This is the third and the last topicof this paper. For some applications (e.g. see a natural connection with Remark 1.6), one studiesthe asymptotic behavior of R X Tr e − t e (cid:3) b,m ( x, x ) dv X ( x ) . More comments on the historical respect comein the beginning of Section 7. Suppose M is an orbifold of (real) dimension k and H ( t, x, x ) is theassociated heat kernel on the diagonal for the standard Laplacian on M . It is known in 2008 [20] (forLaplacian on functions; see also Richardson [57, Theorem 3.5]) that(1.30) Z M H ( t, x, x ) dv X ( x ) ∼ t − k a k + t − k + a k − + t − k +1 a k − + t − k + a k − + · · · , where a s ∈ R is independent of t , s = k, k − , k − , . . . . A novelty is that apart from the overall t − k the expansion is a power series in t .By a strategy partly in connection with the proof of Theorem 1.3, we obtain an expansion of thetrace integral similar to (1.30) in spirit. We find that in our case, the expansion is a power series stillin integral power of t . However, there appear various corrections (depending on m ) supported on eachstratum (cf. (7.49) and (7.54)) in contrast to the expansion in the globally free case (of S action).More precisely, we have (see Theorems 7.20 and 7.24 for more information and proof): Theorem 1.14. (cf. Theorems 7.20, 7.24) With notations in Theorem 1.3 and assumption that the S action is locally free but not globally free, let e be the number (which is even) defined to be the minimumof the (real) codimensions of connected components M of X p ℓ for all ℓ ≥ . For s = n, n − , . . . , we have (1.31) Z X Tr a s,m ( t, x ) dv X ( x ) ∼ q s, + tq s, + t q s, . . . as t → + , where a s,m ( t, x ) (= a s ( t, x )) is as in (1.17) and q s,j ∈ R is independent of t (dependent on m though), j = 0 , , , · · · . Similarly, as t → + , Z X Tr e − t e (cid:3) b,m ( x, x ) dv X ( x ) ∼ ( p X s =1 e i π ( s − p m ) (cid:0) t − n c n + t − n +1 c n − + t − n +2 c n − + · · · (cid:1) + t − n + e ˜ c n − e + O ( t − n + e +1 ) . (1.32) These coefficients satisfy the following. For an ℓ ≥ , write { M ℓ,γ ℓ } γ ℓ (possibly empty for some ℓ )for those connected components M ℓ,γ ℓ of X p ℓ with the codimension codim M ℓ,γ ℓ = e . Set S ℓ,γ ℓ ,s,m = R M ℓ,γℓ Tr α s,m dv M ℓ,γℓ where α s,m (= α s ) is as in (1.18) and the numerical factor (1.33) D ℓ,m = ( √ π ) e X c,h ∈ N , ( h,c )=1 c> ,c | p ℓ ,c p e − i πhc m (cid:12)(cid:12)(cid:12) e i πhc p − (cid:12)(cid:12)(cid:12) e ( > if p ℓ | m ) . i) q s, = q s, = · · · = q s, e − = 0 , q s, = ( p P s =1 e i π ( s − p m ) R X Tr α s,m dv X ( s = n, , n − , n − , . . . ) .ii) q s, e is (a finite sum) of the form P ℓ,γ ℓ D ℓ,m S ℓ,γ ℓ ,s,m ( s = n, , n − , n − , . . . ) .iii) c s = R X Tr α s,m dv X ( s = n, , n − , n − , . . . ) .iv) ˜ c n − e = (2 π ) − ( n +1) P ℓ,γ ℓ D ℓ,m vol ( M ℓ,γ ℓ ) , ( vol = volume ), which is > if p ℓ | m for each ℓ here. The Laplacian in the work [20] is limited to the Laplacian acting on functions while ours aboveis not. We remark that in [57, Theorem 3.5] the nontrivial fractional power in t does occur. Thisis however due partly to a fixed point set of codimension under a reflection isometry ( loc. cit. , p.2315). In our CR case, all of the various fixed point submanifolds are of even (real) codimension, cf.i) of Remark 7.22 or [57, p. 2324]. See Section 7 for a comparison of these methods and results.It will be of interest to study the geometrical significance of the various coefficients in (1.31) and(1.32) as usually studied in the standard heat kernel case. Explicit expressions for more in this regardare available by our treatment, e.g. (7.49), (7.54) and Theorems 7.20, 7.24. EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 15
Remark that the above results essentially deal with the Gaussian part of the heat kernel, whichbehaves as a Dirac type delta function supported on (each) stratum. By contrast, the CR local indextheorem as Corollary 1.13 is derived by exploring the non-Gaussian parts of the heat kernel such asthe off-diagonal estimate in Theorem 5.9. In spirit, the two approaches are complementary to eachother in the present paper, and jointly enhance the understanding of heat kernels for this special classof CR manifolds.Two more remarks go as follows.
Remark . We note that the topological obstruction exists for a CR manifold to admit a transversalCR S action. For instance, a compact strictly pseudoconvex CR -manifold must have even firstBetti number if admitting a transversal CR S action. The reason is that such a manifold must bepseudohermitian torsion free (see [47]), and this vanishing pseudohermitian torsion implies even firstBetti number as shown by Alan Weinstein (see the Appendix in [16]). In this paper, we only considerthe S action that is transversal and locally free. Here are two examples:Example I: X = n ( z , z , z ) ∈ C ; | z | + | z | + | z | + (cid:12)(cid:12) z + z (cid:12)(cid:12) + (cid:12)(cid:12) z + z (cid:12)(cid:12) = 1 o . Then X admitsa transversal CR locally free S action: e − iθ ◦ ( z , z , z ) = ( e − iθ z , e − iθ z , e − iθ z ) . It is clear that this S action is not globally free.Example II: Let X be a compact orientable Seifert -manifold. Kamishima and Tsuboi [44] provedthat X is a compact CR manifold with a transversal CR locally free S action. X is S -fibered over apossibly singular base (an orbifold).In Section 1.5, we collect more examples. Remark . The S action might admit a reduction to a simpler one as Hom( S , S ) = id . Recall p = p < p < p < · · · < p k , associated with periods of X under the given S action ( e − iθ , x ) → e − iθ ◦ x . Then p = p divides each p j , j > . For, the isotropy subgroup Z p ( = Z /p Z ) ⊂ S acts triviallyon the principal stratum, which is dense and open, hence on the whole X by continuity. The isotropysubgroups Z p j , j = 2 , . . . , k , on any other stratum must contain Z p , giving p j p ∈ N .One renormalizes the given S action by the new S action satisfying p = 1 . More precisely, define S × X → X, ( e − iθ , x ) → e − iθ ⋄ x := e − i θp ◦ x. The new S -action ( e − iθ , ⋄ ) has p = 1 . Let e ω be the global real one form with respect to ( e − iθ , ⋄ ) andlet e H qb,m ( X, E ) be the corrsponding cohomology group with respect to ( e − iθ , ⋄ ) . One sees e ω = pω , e H qb,m ( X, E ) = H qb,pm ( X, E ) , ∀ m ∈ Z , ∀ q = 0 , , , . . . , n. (1.34)Examining (1.34) and Corollary 1.13 yields that the index formulas in both cases can be transformedto each other.1.3. Applications.
Applications in CR geometry.
In CR geometry, it has been an important issue to produce manyCR functions or CR sections. Put H b ( X, E ) = (cid:8) u ∈ C ∞ ( X, E ); ∂ b u = 0 (cid:9) . The following belongs to one of the standard questions in this respect.
Question . Let X be a compact weakly pseudoconvex CR manifold. When is the space H b ( X, E ) large? (Pseudoconvex CR manifolds will be briefly reviewed following Definition 2.2.) EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 16
In [48] Lempert proved that a three dimensional compact strongly pseudoconvex CR manifold X with a transversal CR locally free S action can be CR embedded into C N . In [25] Epstein proved thata three dimensional compact strongly pseudoconvex CR manifold X with a transversal CR globallyfree S action can be embedded into C N by the positive Fourier components.The embeddability of X by positive Fourier coefficients is related to the behavior of the S actionon X . For example, suppose for f , . . . , f d m ∈ H b,m ( X ) and g . . . , g h l ∈ H b,l ( X ) the map Φ m,l : x ∈ X → ( f ( x ) , . . . , f d m ( x ) , g ( x ) , . . . , g h l ( x )) ∈ C d m + h l is a CR embedding. Then, the S action on X naturally induces an S action on Φ m,l ( X ) , given by thefollowing:(1.35) e − iθ ◦ ( z , . . . , z d m , z d m +1 , . . . , z d m + h l ) = ( e − imθ z , . . . , e − imθ z d m , e − ilθ z d m +1 , . . . , e − ilθ z d m + h l ) . In short, under a CR embedding by positive Fourier components, one can describe the S actionexplicitly. Conversely, to study the embedding theorem of those CR manifolds by positive Fouriercomponents, it becomes important to know Question . When is dim H b,m ( X, E ) ≈ m n for m large?We shall answer, combining our index theorems with some vanishing theorems (see below), Ques-tion 1.17 and Question 1.18 for CR manifolds with transversal CR locally free S action.Firstly it follows from Corollary 1.13 (by extracting the leading coefficient of the term m n ) Corollary 1.19.
In the same assumption as in Corollary 1.13, one has n X j =0 ( − j dim H jb,m ( X, E )= r ( p X s =1 e π ( s − p mi ) m n n !(2 π ) n +1 Z X ( − dω ) n ∧ ω + O ( m n − ) , (1.36) where r denotes the complex rank of the vector bundle E . For a vanishing theorem we can repeat the proof of Theorem 2.1 in [43] with minor change and get
Proposition 1.20.
In the same assumption as in Corollary 1.13, suppose further that X is weakly pseu-doconvex. Then, for m ≫ H jb,m ( X, E ) = o ( m n ) , for every j = 1 , , . . . , n . Combining Corollary 1.19 and Proposition 1.20 one has
Corollary 1.21.
In the same assumption as in Proposition 1.20 (with X being weakly pseudoconvex).One has, for m ≫ , dim H b,m ( X, E )= r ( p X s =1 e π ( s − p mi ) m n n !(2 π ) n +1 Z X ( − dω ) n ∧ ω + o ( m n ) , where r denotes the complex rank of the vector bundle E . In particular, if the Levi form is stronglypseudoconvex at some point of X , then dim H b,pm ( X ) ≈ m n for m ≫ , and hence dim H b ( X, E ) = ∞ . These results have provided answers to Question 1.17 and Question 1.18 (for our class of CR man-ifolds).For another application, it is of great interest in CR geometry to study whether and when a CRmanifold X can be CR embedded into a complex space. It is a classical theorem of L. Boutet deMonvel [8] which asserts that X can be globally CR embedded into C N for some N ∈ N provided that X is compact (with no boundary), strongly pseudoconvex, and of dimension greater than or equal tofive. EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 17
When X is not strongly pseudoconvex, the space of global CR functions could even be trivial. Asmany interesting examples live in the projective space (e. g. the quadric { [ z ] ∈ CP N − ; | z | + . . . + | z q | − | z q +1 | − . . . − | z N | = 0 } ), it is natural to consider a setting analogous to the Kodaira embeddingtheorem and ask if X can be embedded into the projective space by means of CR sections of a CR linebundle L → X or its k -th power L k .For a study into the above question it is natural to seek the case where the dimension of the space H b ( X, L k ) of CR sections of L k is large as k → ∞ (so one may hopefully find many CR sections tocarry out the embedding). In this regard the following question is asked by Henkin and Marinescu [50,p.47-48]. Question . When is dim H b ( X, L k ) ≈ k n +1 for k large?Assume that L is a rigid CR line bundle with a rigid
Hermitian fiber metric h L (i.e. L a CR linebundle admitting a compatible S action, cf. the beginning of Section 1.2). Let R L ∈ Ω ( X ) be thecurvature of L associated to h L . For a local trivializing ( S -invariant) section s of L , | s ( x ) | h L = e − φ ( x ) with T φ = 0 . Then R L = 2 ∂ b ∂ b φ ∈ Ω ( X ) . ( L k , h L k ) denotes the k -th power of ( L, h L ) .With Corollary 1.13 one can show Proposition 1.23.
With the notations above, for k large we have X m ∈ Z , | m |≤ kδ n X j =0 ( − j dim H jb,m ( X, L k ⊗ E )= r (2 π ) − n − n ! k n +1 Z X Z [ − δ,δ ] ( i R Lx − sdω ( x )) n ∧ ω ( x ) ds + o ( k n +1 ) , (1.37) where δ > and r denotes the complex rank of the vector bundle E .Proof. By Corollary 1.13 one can check X m ∈ Z , | m |≤ kδ n X j =0 ( − j dim H jb,m ( X, L k ⊗ E )= r (2 π ) − n − X m ∈ Z , | m |≤ kδ ( p X s =1 e π ( s − p mi ) 1 n ! Z X ( ik R Lx − mdω ( x )) n ∧ ω ( x ) + o ( k n )= r (2 π ) − n − X m ∈ Z , | m |≤ kδ ( p X s =1 e π ( s − p mi ) k n n ! Z X ( i R Lx − mk dω ( x )) n ∧ ω ( x ) + o ( k n ) . (1.38)Note P ps =1 e π ( s − p mi = p if p | m , and otherwise. By this and (1.38) we get X m ∈ Z , | m |≤ kδ n X j =0 ( − j dim H jb,m ( X, L k ⊗ E )= r (2 π ) − n − X ℓ ∈ Z , | pℓ |≤ kδ pn ! k n Z X ( i R Lx − pℓk dω ( x )) n ∧ ω ( x ) + o ( k n ) . (1.39)It is clear that the (Riemann) sum P ℓ ∈ Z , | pℓ |≤ kδ pk R X ( i R Lx − pℓk dω ( x )) n ∧ ω ( x ) converges to Z X Z [ − δ,δ ] ( i R Lx − sdω ( x )) n ∧ ω ( x ) ds EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 18 as k → ∞ . Hence X ℓ ∈ Z , | pℓ |≤ kδ { pn ! k n Z X ( i R Lx − pℓk dω ( x )) n ∧ ω ( x ) } + o ( k n )= { n ! k n +1 Z X Z [ − δ,δ ] ( i R Lx − sdω ( x )) n ∧ ω ( x ) ds } + o ( k n +1 ) . (1.40)Combining (1.40) with (1.39) we have (1.37). (cid:3) The following two results may be viewed as a companion of the Grauert-Riemenschneider criterionin the CR case (with S action). To start with Definition 1.24.
We say that ( L, h L ) is positive at p ∈ X if the curvature R Lp is a positive Hermitianquadratic form over T , p X . We say that ( L, h L ) is semipositive if for any x ∈ X there exists a constant δ > such that R Lx − sidω ( x ) is a semipositive Hermitian quadratic form over T , x X for any | s | < δ. We can repeat the proof of Theorem 1.24 in [42] with minor change and get
Proposition 1.25. (Asymptotical vanishing)
Assume that ( L, h L ) is a semi-positive CR line bundle over X . Then, for δ > , δ small, we have X m ∈ Z , | m |≤ kδ dim H jb,m ( X, L k ⊗ E ) = o ( k n +1 ) , j = 1 , , . . . , n. Combining Proposition 1.23 and Proposition 1.25, we get
Corollary 1.26. (Bigness)
Assume that ( L, h L ) is semi-positive. Then, for δ > , δ small, we have X m ∈ Z , | m |≤ kδ dim H b,m ( X, L k ⊗ E )= r (2 π ) − n − n ! k n +1 Z X Z [ − δ,δ ] ( i R Lx − sdω ( x )) n ∧ ω ( x ) ds + o ( k n +1 ) , (1.41) where r denotes the complex rank of the vector bundle E . In particular, if ( L, h L ) is positive at some pointof X , then dim H b,m ( X, L k ⊗ E ) ≈ k n +1 . The above result yields an answer to Question 1.22 in the case pertinent to our class of CR mani-folds.1.4.
Kawasaki’s Hirzebruch-Riemann-Roch and Grauert-Riemenschneider criterion for orbifoldline bundles.
There is a link between our CR result and a complex geometry result of Kawasaki onHirzebruch-Riemann-Roch formula over complex orbifolds [46]. Compared to Kawasaki’s, we get asimpler Hirzebruch-Riemann-Roch formula for some class of orbifold line bundles using our secondmain result Corollary 1.13. Moreover, from Corollary 1.21 it follows a Grauert-Riemenschneider cri-terion for orbifold line bundles.To the aim we shortly review the orbifold geometry and also set up notations. Let M be a manifoldand G a compact Lie group. Assume that M admits a G -action: G × M → M, ( g, x ) → g ◦ x. We suppose that the action G on M is locally free, that is, for every point x ∈ M , the stabilizer group G x = { g ∈ G ; g ◦ x = x } of x is a finite subgroup of G . In this case the quotient space(1.42) M/G
EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 19 is known to be an orbifold. A remark of Kawasaki [45, p. 76] discusses the validity of a conversestatement about when a space has a presentation of the form (1.42). (As is well-known, these spacesare actually called V-manifolds by Satake [58], a slightly restrictive class of orbifolds.)We assume now that M is a compact connected complex manifold with complex structure T , M . G induces an action on C T M : G × C T M → C T M, ( g, u ) → g ∗ u, (1.43)where g ∗ = ( g − ) ∗ the push-forward by g − on C T M . Suppose G acts holomorphically, that is g ∗ ( T , M ) ⊂ T , M for g ∈ G . Put C T ( M/G ) := C T M/G , T , ( M/G ) := T , M/G and T , ( M/G ) := T , M/G . Assume that T , ( M/G ) T T , ( M/G ) = { } . Then, T , ( M/G ) gives a complex structureon M/G and
M/G becomes a complex orbifold. Suppose dim C T . ( M/G ) = n .Let L be a G -invariant holomorphc line bundle over M , that is, there exists a choice of transitionfunctions h (defined on open charts U ) of L such that h ( g ◦ x ) = h ( x ) for every g ∈ G , x ∈ U with g ◦ x ∈ U . Suppose that L admits a locally free G -action compatible with that on M , i.e. an action ( g, v ) ( ∈ G × L ) → g ◦ v ∈ L with the property π ( g ◦ v ) = g ◦ ( π ( v )) ( g linearly acts on fibers of L ), π : L → M the projection. Then, L/G is an orbifold holomorphic line bundle over
M/G (the fiber isnot necessarily a vector space).The above construction induced by (locally free) G -action on L naturally extends to L m , the m -thpower of L , and L ∗ , the dual line bundle of L . Thus L m /G and L ∗ /G are also orbifold holomorphicline bundles over M/G . Put ( q = 0 , , , . . . , n )(1.44) Ω ,q ( M/G, L m /G ) := (cid:8) u ∈ Ω ,q ( M, L m ); g ∗ u = u, ∀ g ∈ G (cid:9) . The Cauchy-Riemann operator ∂ : Ω ,q ( M, L m ) → Ω ,q +1 ( M, L m ) is G -invariant, hence gives a ∂ -complex ( ∂, Ω , • ( M/G, L m /G )) and the q -th Dolbeault cohomology group: H q ( M/G, L m /G ) := Ker ∂ : Ω ,q ( M/G, L m /G ) → Ω ,q +1 ( M/G, L m /G )Im ∂ : Ω ,q − ( M/G, L m /G ) → Ω ,q ( M/G, L m /G ) . Let
Tot ( L ∗ ) be the space of all non-zero vectors of L ∗ . Assume that Tot ( L ∗ ) /G is a smooth man-ifold. Take any G -invariant Hermitian fiber metric h L ∗ on L ∗ , set e X = { v ∈ L ∗ ; | v | h L ∗ = 1 } and put X = e X/G . Since
Tot ( L ∗ ) /G is a smooth manifold by the foregoing assumption, X = e X/G is asmooth manifold. The natural S action on e X induces a locally free S action e − iθ on X . One cancheck that X is a CR manifold and the S action on X is CR and transversal.In a similar vein as the proof of Theorem 1.2 one can show (for q = 0 , , , . . . , n and m ∈ Z )(1.45) H q ( M/G, L m /G )) ∼ = H qb,m ( X ) . We pause and introduce some notations. For every x ∈ Tot ( L ∗ ) and g ∈ G , put N ( g, x ) = 1 if g / ∈ G x and N ( g, x ) = inf (cid:8) ℓ ∈ N ; g ℓ = id (cid:9) if g ∈ G x . Set(1.46) p = inf { N ( g, x ); x ∈ Tot ( L ∗ ) , g ∈ G, g = id } . Putting together Corollary 1.13 and (1.45) gives
Theorem 1.27.
With the notations above, recall that we work with assumptions that M is connected and Tot ( L ∗ ) /G is smooth. Then (for every m ∈ Z ) n X j =0 ( − j dim H j ( M/G, L m /G )= ( p X s =1 e π ( s − p mi ) 12 π Z X Td b ( T , X ) ∧ e − m dω π ∧ ω . (1.47) EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 20
To compare this result with that of Kawasaki ([46]) we assume p = 1 for simplicity. Note X is smooth (yet M/G could be singular). The above integral (1.47) reduces to an integral over theprincipal stratum of
M/G (by integrating ω along the fiber S , which gives ). It is thus the sameas to say that the contributions from the lower dimensional strata sum up to zero . As remarked inIntroduction a vanishing result as such is not readily available in the formula of [46]. Note that thenotion of “orbifold” in [46] is slightly more general than that of Satake (on which the present sectionis based). As this generality does no real harm to the reasoning above, we omit the details in thisregard.The above result on the vanishing of the contributions from strata may also be reflected in the indexformula of the works [55], [33] which study the index of transversally elliptic operators on a smoothcompact manifold with the action of a compact Lie group G , by using the framework of equivariantcohomology theory. A remarkable point is that they define the index as a generalized function (on G )(also discussed in Atiyah [1], p.9-17). In fact it is not difficult to verify that for the case of the S action, our present m -th index is basically the m -th Fourier component of the corresponding index (inthe sense of generalized functions) of theirs (for the case g = id ∈ G in [33], [55]).The consistency of our result with those works above helps to shape our own view towards theasymptotic expansion of a (transversal) heat kernel conceived in this subject.For examples that satisfy Theorem 1.27 we refer to Section 1.5. There, we construct, among others,an orbifold holomorphic line bundle over a singular complex orbifold such that the assumptions ofTheorem 1.27 and Corollary 1.28 are fulfilled (see Corollary 1.30 and Subsection 1.5.2 below). Indeedthere are ample examples in this respect.As promised in the beginning of this section, we obtain now a Grauert-Riemenschneider criterionfor orbifold line bundles, upon combining Corollary 1.21 and (1.45). Corollary 1.28.
With the notations and assumptions above, suppose that there is a G -invariant Hermit-ian fiber metric h L on L such that the associated curvature R L is semipositive and positive at some pointof M . Then dim H ( M/G, L pm /G ) ≈ m n for m ≫ and p as in (1.46) . Examples.
In this subsection, some examples of CR manifolds with locally free S action (in-cluding those fitting Theorem 1.27 above) are collected.We first review the construction of generalized Hopf manifolds introduced by Brieskorn and Van deVen [10].1.5.1. Generalized Hopf manifolds.
Let a = ( a , . . . , a n +2 ) ∈ N n +2 , let z = ( z , . . . , z n +2 ) be the stan-dard coordinates of C n +2 and let M ( a ) be the affine algebraic variety given by the equation n +2 X j =1 z a j j = 0 . If some a j = 1 , the variety M ( a ) is non-singular. Otherwise M ( a ) has exactly one singular point,namely , . . . , . Put ^ M ( a ) := M ( a ) − { } . Now we define a holomorphic C -action on ^ M ( a ) by t ◦ ( z , . . . , z n +2 ) = ( e ta z , . . . , e tan +2 z n +2 ) , t ∈ C , ( z , . . . , z n +2 ) ∈ ^ M ( a ) . It is easy to see that the Z -action on ^ M ( a ) is globally free. The equivalence class of ( z , . . . , z n +2 ) ∈ C n +2 with respect to the Z -action is denoted by ( z , . . . , z n +2 ) + Z and hence H ( a ) := ^ M ( a ) / Z = n ( z , . . . , z n +2 ) + Z ; ( z , . . . , z n +2 ) ∈ ^ M ( a ) o is a compact complex manifold of complex dimension n +1 . We call H ( a ) a (generalized) Hopf manifold .Let Γ a be the discrete subgroup of C , generated by and παi , where α is the least common multipleof a , a , . . . , a n +2 . Consider the complex -torus T a = C / Γ a . H ( a ) admits a natural T a -action. Put EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 21 V ( a ) := H ( a ) /T a . By Holmann [35], V ( a ) is a complex orbifold. Let π a : H ( a ) → V ( a ) be the naturalprojection.The following is well-known (see the discussion before Proposition 4 in [10]). Theorem 1.29.
Let p = ( z , . . . , z n +2 ) + Z ∈ H ( a ) . Assume that there are exactly k coordinates z j , . . . , z j k all different from zero, k ≥ . Then, V ( a ) is non-singular at π a ( p ) if and only if [ a , . . . , a n +2 ][ a j , · · · , a j k ] = Y ℓ/ ∈{ j ,...,j k } [ a , . . . , a n +2 ][ a , . . . , a ℓ − , a ℓ +1 , · · · , a n +2 ] . where [ m , . . . , m d ] denotes the least common multiple of m , . . . , m d ∈ N . It follows readily
Corollary 1.30.
Assume n ≥ and ( a , a , . . . , a n +2 ) = (4 b , b , b , b , . . . , b n +2 ) , where b j ∈ Z , b j is odd, j = 1 , . . . , n + 2 . Let p = (0 , , , i, , , . . . ,
0) + Z ∈ H ( a ) . Then, V ( a ) is singular at π a ( p ) . The ideas in the next two (sub)subsections are heavily based on Theorem 1.29 and Corollary 1.30.1.5.2.
Smooth orbifold circle bundle over a singular orbifold.
Put X := { ( z , . . . , z n +2 ) ∈ C n +2 ; z a + z a + · · · + z a n +2 n +2 = 0 , | z | a + | z | a + | z | a + · · · + | z n +2 | a n +2 = 1 } . (1.48)It can be checked that X is a compact weakly pseudoconvex CR manifold of dimension n + 1 withCR structure T , X := T , C n +2 T C T X , where T , C n +2 denotes the standard complex structure on C n +2 .Let α be the least common multiple of a , . . . , a n +2 . Consider the following S action on X : S × X → X,e − iθ ◦ ( z , . . . , z n +2 ) → ( e − i αa θ z , . . . , e − i αan +2 θ z n +2 ) . (1.49)One sees that the S action is well-defined, locally free, CR and transversal. Moreover one has thatthe quotient X/S is equal to V ( a ) , a = ( a , a , · · · , a n +2 ) . Hence, X/S is a complex orbifold.One sees, by using Corollary 1.30, that the above X/S is singular if n ≥ and ( a , . . . , a n +2 ) =(4 b , b , b , b , . . . , b n +2 ) , where b j ∈ Z , b j is odd, j = 1 , , . . . , n + 2 .We now show that ( X, T , X ) is CR-isomorphic to the (orbifold) circle bundle associated with anorbifold line bundle over X/S = V ( a ) .To see this and to construct the circle bundle in the first place, let L = ( ^ M ( a ) × C ) / ≡ , where ( z , . . . , z n +2 , λ ) ≡ ( e z , . . . , e z n +2 , e λ ) if e z j = e maj z j , j = 1 , . . . , n + 2 , e λ = e m λ, where m ∈ Z . We can check that ≡ is an equivalence relation and L is a holomorphic line bundleover H ( a ) . The equivalence class of ( z , . . . , z n +2 , λ ) ∈ ^ M ( a ) × C is denoted by [( z , . . . , z n +2 , λ )] . Thecomplex -torus T a action on L is given by the following: T a × L → L, ( t + iθ ) ◦ [( z , . . . , z n +2 , λ )] → [( e t + iθa z , . . . , e t + iθan +2 z n +2 , e t − i θα λ )] , (1.50)where α is the least common multiple of a , . . . , a n +2 . One has that the torus action (1.50) is well-defined and L/T a is an orbifold line bundle over H ( a ) /T a = V ( a ) . EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 22
Let τ : L → L/T a be the natural projection and for [( z , . . . , z n +2 , λ )] ∈ L , we write τ ([ z , . . . , z n +2 , λ ]) =[ z , . . . , z n +2 , λ ] + T a . One sees that the pointwise norm (cid:12)(cid:12) [( z , . . . , z n +2 , λ )] + T a (cid:12)(cid:12) h L/Ta := | λ | (cid:16) | z | a + | z | a + | z | a + · · · + | z n +2 | a n +2 (cid:17) − is well-defined as a Hermitian fiber metric on L/T a . The (orbifold) circle bundle C ( L/T a ) with respectto ( L/T a , h L/T a ) is given by C ( L/T a ) : = n v ∈ L/T a ; | v | h L/Ta = 1 o = n [( z , . . . , z n +2 , λ )] + T a ; | λ | = | z | a + | z | a + | z | a + · · · + | z n +2 | a n +2 o . (1.51)One sees that C ( L/T a ) is a smooth CR manifold with the CR structure T , C ( L/T a ) := T , L/T a \ C T C ( L/T a ) , where T , C ( L/T a ) denotes the complex structure on L/T a . Moreover, the orbifold line bundle L/T a → V ( a ) satisfies a similar situation as in Theorem 1.27 (i.e. the space X/S = V ( a ) hereas the M/G there, is singular and C ( L/T a ) as a (orbifold) circle bundle over M/G is smooth).We are ready to give an CR isomorphism of X and the (orbifold) circle bundle C ( L/T a ) . Note C ( L/T a ) admits a nature S action: e − iθ ◦ ([( z , . . . , z n +2 , λ )] + T a ) = [( z , . . . , z n +2 , e − iθ λ )] + T a , Let
Φ : C ( L/T a ) → X be the smooth map defined as follows. For every [( z , . . . , z n +2 , λ )] + T a ∈ C ( L/T a ) , there is a unique (ˆ z , . . . , ˆ z n +2 ) ∈ X such that [( z , . . . , z n +2 , λ )] + T a = [(ˆ z , . . . , ˆ z n +2 , T a . Then,
Φ([( z , . . . , z n +2 , λ )] + T a ) := (ˆ z , . . . , ˆ z n +2 ) ∈ X . It can be checked that Φ is a CR embedding,globally one to one, onto and the inverse Φ − : X → C ( L/T a ) is also a CR embedding. Moreover e − iθ ◦ Φ( x ) = Φ( e − iθ ◦ x ) , ∀ x ∈ C ( L/T a ) . We conclude Φ is a CR isomorphism.1.5.3. Family, non-pseudoconvex cases and deformations.
In the notation of Subsection 1.5.1 we as-sume a = 1 , so M ( a ) = (cid:8) ( z , . . . , z n +2 ) ∈ C n +2 ; z = − z a − · · · − z a n +2 n +2 (cid:9) . Fix a q = 2 , , . . . , n + 1 . Put, for t ∈ C , X q,t := { ( z , . . . , z n +2 ) + Z ∈ H ( a ); − | z a + tz a | − | z | a − · · · − | z q | a q + | z q +1 | a q +1 + · · · + | z n +2 | a n +2 = 0 } . (1.52)One can check that for each t , X q,t is a compact CR manifold of dimension n + 1 with CR structure T , X q,t := T , H ( a ) T C T X q,t , where T , H ( a ) denotes the natural complex structure inherited by M ( a ) . Note X q,t is diffeomorphic to X q,t for t , t ∈ C since they can be connected through a(smooth) family of compact manifolds.Let e a be the least common multiple of a , . . . , a q . Consider the following S action on X q,t : S × X q,t → X q,t ,e − iθ ◦ (( z , . . . , z n +2 ) + Z ) → ( e − i e aθ ( − z a − · · · − z a q q ) − z a q +1 q +1 − · · · − z a n +2 n +2 , e − i e aa θ z , . . . , e − i e aaq θ z q , z q +1 , . . . , z n +2 ) + Z . (1.53)One sees that the S action is well-defined, locally free, CR and transversal. This is an example for afamily of CR manifolds admitting a transversal CR locally free S action.Moreover these CR manifolds X q,t are not pseudoconvex. EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 23
Now we consider certain CR deformations of a compact CR manifold X with a transversal CRlocally free S action. Let F ( x ) ∈ C ∞ ( X ) with T F = 0 ( T the global real vector field induced by the S action). Let Z , . . . , Z n ∈ C ∞ ( X, T , X ) be a basis for T , X . Put(1.54) H , X := { Z j + Z j ( F ) T ; j = 1 , , . . . , n } . One can check that H , X is a CR structure and the S action is locally free, CR and transversal withrespect to this new CR structure H , X (see (2.10) via the BRT construction).To see how “new” this CR structure H , X is, let’s take X to be a circle bundle associated with aholomorphic line bundle ( L, || · || ) over a compact complex manifold M . Consider a change of metric || · || → e − f || · || on L and the circle bundle e X thus induced by this new metric. By using the formula(1.56) below one sees that H , X of (1.54) for F = − if is equivalent to T , e X . But is ( X, T , X ) CR equivalent to ( e X, T , e X ) ? The answer is in general no. For instance, spherical CR structures ona certain topological type of X can be obtained by using special metrics on L (cf. [18]). Hence anarbitrary perturbation of the bundle metric, say by the multiplier e − f , would bring X out of thespherical category. Note that the moduli space of spherical CR structures in [18] is finite dimensional.It follows that for F a purely imaginary function on X , the CR structure H , X is in general not CRequivalent to T , X .If, however, F is a real function, it is easily seen that the change ( z, θ ) → ( z, θ + F ) is globallydefined, hence it gives a diffeomorhism φ of X . One sees φ ∗ ( T , X ) = H , X , cf. (1.56) below. So inthis case the CR structure H , X is equivalent to the original one.1.6. Proof of Theorem 1.2.
Notations as in Theorem 1.2 let s be a local trivializing section of L defined on some open set U of M , | s | h L = e − φ . Let z = ( z , . . . , z n ) be holomorphic coordinates on U . We identify U with an open set of C n and have the local diffeomorphism:(1.55) τ : U × ] − ε , ε [ → X , ( z, θ ) e − φ ( z ) s ∗ ( z ) e − iθ , < ε ≤ π. Put D = U × ] − ε , ε [ as a canonical coordinate patch with ( z, θ ) canonical coordinates (withrespect to the trivialization s ) such that T = ∂∂θ (recall T is the global real vector field induced by the S action). Moreover one has T , X = (cid:26) ∂∂z j − i ∂φ∂z j ( z ) ∂∂θ ; j = 1 , , . . . , n (cid:27) ,T , X = (cid:26) ∂∂z j + i ∂φ∂z j ( z ) ∂∂θ ; j = 1 , , . . . , n (cid:27) , (1.56)and(1.57) T ∗ , X = { dz j ; j = 1 , , . . . , n } , T ∗ , X = { dz j ; j = 1 , , . . . , n } . See also Theorem 2.9 and proof of Proposition 4.2 for similar formulas in the general case of S action.Let f ( z ) ∈ Ω ,q ( D ) . By (1.57) we may identify f with an element in Ω ,q ( U ) .The key object in our proof is the map A ( q ) m : Ω ,qm ( X ) → Ω ,q ( M, L m ) , to be defined as follows. Let u ∈ Ω ,qm ( X ) . We can write u ( z, θ ) = e − imθ ˆ u ( z ) (on D ) for some ˆ u ( z ) ∈ Ω ,q ( U ) . Then, on U ⊂ M , wedefine(1.58) A ( q ) m u := s m ( z ) e mφ ( z ) ˆ u ( z ) ∈ Ω ,q ( U, L m ) . We need to check the following.i) A ( q ) m in (1.58) is well-defined, hence gives rise to a global element A ( q ) m u ∈ Ω ,q ( M, L m ) .ii) It satisfies the commutativity ∂A ( q ) m = A ( q +1) m ∂ b (thus induces a map on respective cohomologies). EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 24
To check i) let s and s be local trivializing sections of L on an open set U . Let ( z, θ ) ∈ C n × R and ( z, η ) ∈ C n × R be canonical coordinates of D with respect to s and s respectively ( D as the above).Set | s | h L = e − φ and | s | h L = e − φ . We write (on D ) u = e − imθ ˆ u ( z ) u = e − imη ˆ u ( z ) . (1.59)To check i) amounts to the following(1.60) s m ( z ) e mφ ( z ) ˆ u ( z ) = s m ( z ) e mφ ( z ) ˆ u ( z ) , ∀ z ∈ U. Let s = gs for g a unit on U . To find relations between φ and φ , ˆ u and ˆ u in terms of g , | s | h L = e − φ = | g | | s | h L = e | g |− φ , giving(1.61) φ = φ − log | g | . For ˆ u and ˆ u , we first claim the following ( τ in (1.55) for ( z, θ ) and τ the similar one for ( z, η ) )(1.62) If τ ( z, θ ) = τ ( z, η ) , then e − iθ (cid:0) g ( z ) g ( z ) (cid:1) = e − iη (with a certain branch of the square root) . Proof of the claim (1.62) . Combining (1.55) and (1.61) one sees τ ( z, θ ) = s ∗ ( z ) e − iθ − φ ( z ) = s ∗ ( z ) g ( z ) e − iθ − φ ( z ) = s ∗ ( z ) g ( z ) e − iθ − φ ( z ) − log | g ( z ) | = s ∗ ( z ) (cid:0) g ( z ) g ( z ) (cid:1) e − iθ − φ ( z ) . (1.63)The condition τ ( z, θ ) = τ ( z, η ) is the same as to say, by (1.55),(1.64) s ∗ ( z ) e − iθ − φ ( z ) = s ∗ ( z ) e − iη − φ ( z ) . By (1.63) and (1.64) we deduce that (cid:0) g ( z ) g ( z ) (cid:1) e − iθ = e − iη , as claimed. (cid:3) Now that the relations (1.61) and (1.62) have been found, the (1.60) follows by using (1.59).Hence A ( q ) m : Ω ,qm ( X ) → Ω ,q ( M, L m ) is well-defined, proving i) above.Moreover it is easily checked that A ( q ) m is bijective. We omit the detail.To prove ii) that ∂A ( q ) m = A ( q +1) m ∂ b , by (1.56) and (1.57) one sees (on D )(1.65) ∂ b u = ∂ b ( e − imθ ˆ u ) = n X j =1 e − imθ dz j ∧ (cid:0) ∂ ˆ u∂z j ( z ) + m ∂φ∂z j ( z )ˆ u ( z ) (cid:1) . Hence (1.65) and (1.58) yield A ( q +1) m ( ∂ b u ) = s m ( z ) e mφ ( z ) n X j =1 dz j ∧ (cid:0) ∂ ˆ u∂z j ( z ) + m ∂φ∂z j ( z )ˆ u ( z ) (cid:1) = s m ( z ) ∂ ( e mφ ( z ) ˆ u ( z )) on U , (1.66)giving ∂A ( q ) m = A ( q +1) m ∂ b . Theorem 1.2 follows. Remark . The map A ( q ) m does not depend on the metrics of the manifolds X and M . In latersections we study the Kohn Laplacian and Kodaira Laplacian on X and M respectively, and try toestablish a link between the two Laplacians (with the aim at the Kohn’s). In this regard we need equip X and M with appropriate metrics so that A ( q ) m thus defined is also compatible with these metrics. Notea localization of this (metrical) construction (cf. Proposition 5.1) paves the way for our subsequentplan in this work. EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 25
Some difficulties (and ways out) for a straightforward generalization of the proof for this specialcase (globally free S action) will be discussed in the subsection below.1.7. The idea of the proofs of Theorem 1.3, Theorem 1.10 and Corollary 1.13.
We will give anoutline of main ideas of some proofs. For the proof of Theorem 1.14, some ideas are outlined inthe beginning of Section 7. We refer to Section 2.2 and Section 2.3 for notations and terminologiesused here. The main technical tool of our method lies in a construction of a heat kernel for the KohnLaplacian associated to the m -th S Fourier component.1.7.1.
Global difficulties.
For simplicity we assume that X is CR K¨ahler (cf. Definition 1.11) without E and h · | · i is induced by a CR K¨ahler form Θ on X . Write ∂ ∗ b for the adjoint of ∂ b with respect to ( · | · ) and ∂ ∗ b,m = ∂ ∗ b : Ω ,q +1 m ( X ) → Ω ,qm ( X ) with Ω , + m ( X ) and Ω , − m ( X ) denoting forms of even andodd degree. Consider D ± b,m := ∂ b,m + ∂ ∗ b,m : Ω , ± m ( X ) → Ω , ∓ m ( X ) , m ∈ Z and let (cid:3) + b,m := D − b,m D + b,m : Ω , + m ( X ) → Ω , + m ( X ) ( (cid:3) − b,m := D + b,m D − b,m similarly).Extending (cid:3) + b,m and (cid:3) − b,m to L , + m ( X ) and L , − m ( X ) ( L -completion), respectively in the standardway, we will show in Theorem 3.5 that Spec (cid:3) ± b,m are discrete subsets of [0 , ∞ [ and Spec (cid:3) ± b,m consistof eigenvalues of (cid:3) ± b,m .For ν ∈ Spec (cid:3) + b,m , let (cid:8) f ν , . . . , f νd ν (cid:9) be an orthonormal frame for the eigenspace of (cid:3) + b,m witheigenvalue ν . Write T ∗ , • X = ⊕ ≤ q ≤ n T ∗ ,q X . e − t (cid:3) + b,m ( x, y ) : T ∗ , • y X → T ∗ , + x X , said to be a heatkernel, is given by (cf. (1.12))(1.67) e − t (cid:3) + b,m ( x, y ) = X ν ∈ Spec (cid:3) + b,m d ν X j =1 e − νt f νj ( x ) ∧ ( f νj ( y )) † . (Similarly we can define e − t (cid:3) − b,m ( x, y ) .)We will show in Corollary 4.8 (see also Remark 4.9) that we have a CR McKean-Singer type formula :for t > ,(1.68) n X j =0 ( − j dim H jb,m ( X ) = Z X (cid:16) Tr e − t (cid:3) + b,m ( x, x ) − Tr e − t (cid:3) − b,m ( x, x ) (cid:17) dv X . By this formula the proof of our index theorem (cf. Corollary 1.13) is reduced to determining thesmall t behavior of the function (cid:16) Tr e − t (cid:3) + b,m ( x, x ) − Tr e − t (cid:3) − b,m ( x, x ) (cid:17) .With the kernel e − t (cid:3) + b,m ( x, y ) there is associated an operator denoted by e − t (cid:3) + b,m : Ω , + ( X ) → Ω , + m ( X ) . Note the domain is set to be the full space Ω , + ( X ) . From (1.67) it follows that the kernelsatisfies a heat equation which is expressed in the following operator form(1.69) ∂e − t (cid:3) + b,m ∂t + (cid:3) + b,m e − t (cid:3) + b,m = 0 and(1.70) e − t (cid:3) + b,m | t =0 = Q + m , where Q + m : L , + ( X ) → L , + m ( X ) is the orthogonal projection.The main difficulty lies in that the initial condition (1.70) is a projection operator rather than an identity operator because we are dealing with part of the L space (i.e. the m -th eigenspaces) ratherthan the whole L space (as in the usual case). In a similar vein, let us quote in a paper of Richardson[56, p. 358]: “A point of difficulty that often arises in this area of research is that the space...is not theset of all sections of any vector bundle, and therefore the usual theory of elliptic operators and heat EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 26 kernels does not apply directly...”. The condition (1.70) eventually leads to the result that the heatkernels e − t (cid:3) ± b,m ( x, y ) do not have the standard expansions (as usually seen).For a better understanding let’s assume that X is a (orbifold) circle bundle of an orbifold linebundle L over a K¨ahler orbifold M (see Section 1.5 for specific examples). As in Theorem 1.2 (seeSection 1.6), one sees bijective maps A ± m : Ω , ± m ( X ) → Ω , ± ( M, L m ) such that A − m ∂ b = ∂A + m . Let (cid:3) + m be the Kodaira Laplacian with values in T ∗ , + M ⊗ L m and let e − t (cid:3) + m bethe associated heat operator. Consider B m ( t ) := ( A + m ) − ◦ e − t (cid:3) + m ◦ A + m . A ± m are metric-independent (ona given X ). To get a link between (cid:3) + b,m and (cid:3) + m it requires, however, a compatible choice of metricson X and M . With this done, one checks that B ′ m ( t ) + (cid:3) + b,m B m ( t ) = 0 and B m (0) = I on Ω , + m ( X ) .But B m ( t ) is not the heat operator e − t (cid:3) + b,m . A trivial reason is that B m ( t ) is defined on Ω , + m ( X ) while e − (cid:3) + b,m is on the whole Ω , + ( X ) . In fact one has(1.71) e − t (cid:3) + b,m = (cid:0) ( A + m ) − ◦ e − t (cid:3) + m ◦ A + m (cid:1) ◦ Q + m = B m ( t ) ◦ Q + m (= Q + m ◦ B m ( t ) ◦ Q + m ) . Let B m ( t, x, y ) be the distribution kernel of B m ( t ) . To emphasize the role played by Q + m in ourconstruction, it is illuminating to note the following (cf. (1.71), (2.2) and (4.17))(1.72) e − t (cid:3) + b,m ( x, y ) = 12 π Z π − π B m ( t, x, e − iu ◦ y ) e − imu du. For x ∈ X p (the principal stratum), from (1.72) and the much better known kernel e − t (cid:3) + m (on theprincipal stratum of M ) it follows(1.73) e − t (cid:3) + b,m ( x, x ) ∼ t − n a + n ( x ) + t − ( n − a + n − ( x ) + · · · . However for x / ∈ X p , by lack of the asymptotic expansion of B m ( t ) (or e − t (cid:3) + m on low dimensionalstrata of M ) it is unclear how one can understand the asymptotic behavior of e − t (cid:3) + b,m ( x, x ) by meansof (1.72). This presents a major deviation from proof of the globally free case (as Theorem 1.2, cf.(1.1), (1.2)).To see more clearly the discrepancy between the two cases (locally free and globally free) we notethat the expansion (1.73) converges only locally uniformly on X p , due to a nontrivial contributioninvolving a “distance function” (see Subsection 1.7.3 for more). In fact the expansion of the form(1.73) which is usually seen, cannot hold here ( globally on X ) (cf. Remark 1.6).It is thus not immediate for one to arrive at a detailed understanding of the (transversal) heat kernelby only using the global argument. Even in the (smooth) orbifold circle bundle case, to understandthe asymptotic behavior of the heat operator e − t (cid:3) + b,m we will still need to work directly on the CRmanifold X instead of M .In this paper we give a construction which is independent of the use of orbifold geometry and ismore adapted to CR geometry as our CR manifold X is not assumed to be an orbifold circle bundleof a complex orbifold. Because of the failure of the global argument as just said, we are now led towork on it locally . The framework for this is BRT trivialization (Section 2.4) which is first treated byBaouendi, Rothschild and Treves [4] in a more general context.1.7.2.
Transition to local situation.
Let B := ( D, ( z, θ ) , ϕ ) be a BRT trivialization (see Theorem 2.9).We write D = U × ] − ε, ε [ , where ε > and U is an open set of C n . Let L → U be a trivial line bundlewith a non-trivial Hermitian fiber metric | | h L = e − ϕ (where ϕ ∈ C ∞ ( D, R ) is as in Theorem 2.9)and ( L m , h L m ) → U be the m -th power of ( L, h L ) . Θ (cf. Definition 1.11, recalling X is CR K¨ahleras assumed for the moment) induces a K¨ahler form Θ U on the complex manifold U . Let h · , · i be theHermitian metric on C T U (associated with Θ U ), inducing together with h L m the L inner product ( · , · ) m on Ω , ∗ ( U, L m ) . EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 27
Let ∂ ∗ ,m : Ω ,q +1 ( U, L m ) → Ω ,q ( U, L m ) be the formal adjoint of ∂ with respect to ( · , · ) m . Put, asthe case of D b,m and (cid:3) b,m , D ± B,m := ∂ + ∂ ∗ ,m : Ω , ± ( U, L m ) → Ω , ∓ ( U, L m ) and (cid:3) + B,m := D − B,m D + B,m :Ω , + ( U, L m ) → Ω , + ( U, L m ) .In Proposition 5.1, by the above choice of metrics in forming Laplacians on two different spaces ( D and U ), we can provide a link between these Laplacians, asserting that(1.74) e − mϕ (cid:3) ± B,m ( e mϕ e u ) = e imθ (cid:3) ± b,m ( u ) where as before, u ∈ Ω , ± m ( X ) can be written (on D ) as u ( z, θ ) = e − imθ e u ( z ) for some e u ( z ) ∈ Ω , ± ( U, L m ) ⊂ Ω , ± ( D, L m ) .Write x = ( z, θ ) , y = ( w, η ) (on D ). With (1.74) one expects that the heat kernel e − t (cid:3) + b,m ( x, y ) locally (on D ) should be(1.75) e − mϕ ( z ) − imθ e − t (cid:3) + B,m ( x, y ) e mϕ ( w )+ imη . Thus one obtains local heat kernels on these BRT charts.We would like to patch them up. Assume that X = D S D S · · · S D N (where D j in a BRTtrivialization B j := ( D j , ( z, θ ) , ϕ j ) ) with D j = U j × ] − δ j , e δ j [ ⊂ C n × R , δ j > , e δ j > , U j is an openset in C n ).Let χ j , e χ j ∈ C ∞ ( D j ) ( j = 1 , , . . . , N ). Put A m ( t ) = N X j =1 χ j ( x ) (cid:16) e − mϕ j ( z ) − imθ e − t (cid:3) + Bj,m ( z, w ) e mϕ j ( w )+ imη (cid:17) e χ j ( y ) , P m ( t ) = A m ( t ) ◦ Q + m . (1.76)It is hoped that P m (0) = Q + m and P ′ m ( t ) + (cid:3) + b,m P m ( t ) is small as t → + for certain χ j , e χ j . Thisis related to asymptotic heat kernel . But as we will see, this standard patch-up construction does notquite work out in our case.In short, we will see that in the locally free case the nice (pointwise) relation (1.74) between Kodairaand Kohn Laplacians does not quite carry over to the global objects: heat kernels, whose mutualrelation is to be seen below by more delicate analysis relevant to the presence of strata beyond theprincipal stratum.1.7.3. Local difficulties.
A necessary condition for P m (0) = Q + m is (cf. Lemma 5.10)(1.77) N X j =1 χ j ( x ) Z π − π e χ j ( w, η ) | w = z dη = 1 . For the cut-off functions χ j , e χ j above, a reasonable choice (adapted to BRT trivializations) is thefollowing (for j = 1 , , . . . , N ):i) χ j ( z, θ ) ∈ C ∞ ( D j ) with P Nj =1 χ j = 1 on X ;ii) τ j ( z ) ∈ C ∞ ( U j ) with τ j ( z ) = 1 if ( z, θ ) ∈ Supp χ j ,iii) σ j ∈ C ∞ (] − δ j , e δ j [) with R ˜ δ j − δ j σ j ( η ) dη = 1 .Set e χ j ( y ) ≡ τ j ( w ) σ j ( η ) . Then χ j ( x ) , e χ j ( y ) satisfy (1.77).One can check P m (0) = Q + m and a little more work shows(1.78) P ′ m ( t ) + (cid:3) + b,m P m ( t ) = R m ( t ) ◦ Q + m where for some k ,(1.79) R m ( t ) = N X j =1 k X ℓ =1 L ℓ,j (cid:16) χ j ( x ) e − mϕ j ( z ) − imθ (cid:17) P ℓ,j (cid:16) e − t (cid:3) + Bj,m ( z, w ) (cid:17) e mϕ j ( w )+ imη e χ j ( y ) , EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 28 L ℓ,j is a partial differential operator of order ≥ and ≤ (for all ℓ, j ) and P ℓ,j is a partial differentialoperator of order acting on x (for all ℓ, j ). Since (cid:12)(cid:12)(cid:12)(cid:12) e − t (cid:3) + Bj,m ( z, w ) (cid:12)(cid:12)(cid:12)(cid:12) ∼ t n e − | z − w | t , there could be termsof the form, say(1.80) P ℓ,j (cid:16) e − t (cid:3) + Bj,m ( z, w ) (cid:17) ∼ t n e − | z − w | t | z − w | t . To require P ′ m ( t ) + (cid:3) + b,m P m ( t ) to be small (as t → + ) we need (by substituting (1.80) into (1.79) toget singular terms in powers of t smooth out):(1.81) L ℓ,j (cid:16) χ j ( x ) e − mϕ j ( z ) − imθ (cid:17) e mϕ j ( w )+ imη e χ j ( y ) = 0 if z is close to w ( | z − w | . √ t ) . Since χ j may not be constant on Supp e χ j (for some j ), it is hard for (1.81) to hold. Despite that inthe usual (elliptic) case a construction of the heat kernel using cut-off functions as above is available,in view that a distance function will appear in our asymptotic expression (cf. (1.85) below) it isunclear whether this type of standard construction can be immediately carried out in our case.It turns out that upon transferring to an adjoint version of the original equation one may bypass theaforementioned difficulty (cf. Lemmas 5.10, 5.11), to which we turn now.For j = 1 , , . . . , N there exists A B j , + ( t, z, w ) ( ∈ C ∞ ( R + × U j × U j , T ∗ , + U ⊠ ( T ∗ , + U ) ∗ ) , cf. Theo-rem 5.6), regarded as an adjoint heat kernel , such that lim t → A B j , + ( t ) = I in D ′ ( U, T ∗ , + U ) ,A ′ B j , + ( t ) u + A B j , + ( t )( (cid:3) + B j ,m u ) = 0 , ∀ u ∈ Ω , +0 ( U ) , ∀ t > , (1.82)and A B j , + ( t, z, w ) admits an asymptotic expansion as t → + (see (5.19)). Put(1.83) H j ( t, x, y ) = χ j ( x ) e − mϕ j ( z ) − imθ A B j , + ( t, z, w ) e mϕ j ( w )+ imη e χ j ( y ) . Also set(1.84) Γ( t ) := N X j =1 H j ( t ) ◦ Q + m : Ω , + ( X ) → Ω , + ( X ) . By using the adjoint equation, we can avoid the difficulty mentioned in (1.81) so that Γ( t ) givesan asymptotic (adjoint) heat kernel (see that below (1.76)). To get back to the kernel of the origi-nal equation, we can now start with the adjoint of Γ( t ) . By carrying out the (standard) method ofsuccessive approximation, we can reach the global kernel of the adjoint of (the adjoint of) e − t (cid:3) + b,m (Section 5.2). This yields the kernel of e − t (cid:3) + b,m since e − t (cid:3) + b,m is self-adjoint. More precisely we canprove that (see Theorem 5.14 and Theorem 6.1) (cid:13)(cid:13)(cid:13) e − t (cid:3) + b,m ( x, y ) − Γ( t, x, y ) (cid:13)(cid:13)(cid:13) C ( X × X ) ≤ e − ε t , ∀ t ∈ (0 , ε ) , Γ( t, x, x ) ∼ (cid:16) p X s =1 e π ( s − p mi (cid:17) ∞ X j =0 t − n + j α + n − j ( x ) mod O (cid:16) t − n e − ε d ( x,X sing )2 t (cid:17) , ∀ x ∈ X p , (1.85)where α + s ( x ) ∈ C ∞ ( X, End ( T ∗ , + X ⊗ E )) , s = n, n − , . . . , ε , ε , ε > some constants and ˆ d a sortof “distance function” (discussed above Theorem 1.3).The appearance of this distance function ˆ d may be attributed to the use of projection Q + m in (1.84)(which picks up the m -th Fourier component; see (5.40) and (6.8)). See below for more about thispoint. By the first inequality in (1.85) one obtains the (same) asymptotic expansion(1.86) e − t (cid:3) + b,m ( x, x ) ∼ (cid:16) p X s =1 e π ( s − p mi (cid:17) ∞ X j =0 t − n + j α + n − j ( x ) mod O (cid:16) t − n e − ε d ( x,X sing )2 t (cid:17) EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 29 on X p . Similar results hold for e − t (cid:3) − b,m ( x, x ) .The terms involved in O (cid:16) t − n e − ε d ( x,X sing )2 t (cid:17) of (1.86) are singular (due to t − n as x → X sing ). Onlyupon taking the supertrace can these terms be (partially) cancelled ( t − n dropping out). That is, for x ∈ X p Tr e − t (cid:3) + b,m ( x, x ) − Tr e − t (cid:3) − b,m ( x, x ) ∼ (cid:16) p X s =1 e π ( s − p mi (cid:17) ∞ X j =0 t − n + j (cid:16) Tr α + n − j ( x ) − Tr α − n − j ( x ) (cid:17) mod O (cid:16) e − ε d ( x,X sing )2 t (cid:17) . (1.87)To see this conceptually, let’s take, for instance, (1.71) and (1.72) in which along the diagonal (i.e.setting x = y to the left of (1.72)), the off-diagonal contribution (in the term to the right of the sameequation) still enters nontrivially (unseen in the usual elliptic case) due to the projection Q + m .To get estimates on these off-diagonal terms our argument (cf. Theorem 5.9) is based on therescaling technique of Getzler and on a supertrace identity in Berenzin integral (cf. Prop. 3.21 of [6]),which combine to give the needed (partial) cancellation.From (1.68), (1.86) and (1.87) it follows(1.88) n X j =0 ( − j dim H jb,m ( X, E ) = (cid:16) p X s =1 e π ( s − p mi (cid:17) lim t → + Z X n X ℓ =0 t − ℓ (cid:16) Tr α + ℓ ( x ) − Tr α − ℓ ( x ) (cid:17) dv X ( x ) . Remark that we have had a (transversal) heat kernel which is put in the disguise of the spectral ge-ometry (1.67), (4.15). To our knowledge no argument in the literature claims that (in the transversallyelliptic case) the spectral heat kernel shall have the asymptotic estimates as (1.86). The somewhatlengthy part of our reconstruction of the (transversal) heat kernel (beyond its spectral realization)becomes indispensable as far as our purpose is concerned.1.7.4.
Completion by evaluating local density and by using
Spin c structure. As above we first treat thecase that X is CR K¨ahler (Definition 1.11). In view of (1.88), to complete the proof of our indextheorem (cf. Corollary 1.13) amounts to understanding the small t behavior of the local density n X ℓ =0 t − ℓ (cid:16) Tr α + ℓ ( x ) − Tr α − ℓ ( x ) (cid:17) . Let’s be back to the local situation. Fix x ∈ X p . Let B j = ( D j , ( z, θ ) , ϕ j ) ( j = 1 , , . . . , N ) be BRTtrivializations as before. Assume that x ∈ D j and x = ( z j , ∈ U j ⊂ D j .As our heat kernel (on X ) is related to the local heat kernel (on U j ), one sees (for some N ( n ) ≥ n ) N ( n ) X ℓ =0 t − ℓ (cid:16) Tr α + ℓ ( x ) − Tr α − ℓ ( x ) (cid:17) = 12 π N X j =1 χ j ( x ) (cid:16) Tr A B j , + ( t, z j , z j ) − Tr A B j , − ( t, z j , z j ) (cid:17) + O ( t ) , (1.89)where A B j , + ( t, z, w ) is as in (1.82).By borrowing the rescaling technique in [6] and [23] we can show (in a fairly standard manner, cf.Theorem 5.8 or the second half of this section) that for each j = 1 , , . . . , N , (cid:16) Tr A B j , + ( t, z, z ) − Tr A B j , − ( t, z, z ) (cid:17) dv U j ( z )= [Td ( ∇ T , U j , T , U j ) ∧ ch ( ∇ L m , L m )] n ( z ) + O ( t ) , ∀ z ∈ U j , (1.90)( dv U j the induced volume form on U j ) where Td ( ∇ T , U j , T , U j ) and ch ( ∇ L m , L m ) denote the rep-resentatives of the Todd class of T , U j and the Chern character of L m , respectively. EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 30
A novelty here is Section 2.3 in which we will introduce tangential characteristic classes , tangentialChern character and tangential Todd class on CR manifolds with S action, so that(1.91) [Td ( ∇ T , U j , T , U j ) ∧ ch ( ∇ L m , L m )] n ( z j ) dv U j ( z j ) = [Td b ( ∇ T , X , T , X ) ∧ e − m dω π ∧ ω ] n +1 ( x ) dv X ( x ) , where Td b ( ∇ T , X , T , X ) denotes the representative of the tangential Todd class of T , X (associ-ated with the given Hermitian metric (2.9)). From (1.89), (1.90) and (1.91) it follows n X ℓ =0 t − ℓ (cid:16) Tr α + ℓ ( x ) − Tr α − ℓ ( x ) (cid:17) dv X ( x )= 12 π h Td b ( ∇ T , X , T , X ) ∧ e − m dω π ∧ ω i n +1 ( x ) + O ( t ) , ∀ x ∈ X p . (1.92)(The O ( t ) term to the rightmost of (1.92) actually vanishes by using (5.20).)Combining (1.92) and (1.88) we get our index theorem (cf. Corollary 1.13) when X is CR K¨ahler.When X is not CR K¨ahler , we still have (1.85), (1.86) and (1.88). The ensuing obstackle is more orless known:i) the rescaling technique does not quite work well as the local operator (cid:3) + B j ,m in (1.82) is not goingto be of Dirac type (in a strict sense);ii) it is obscure to understand the small t behavior of A B j , + ( t, z, z ) in this case;iii) (1.90) is not even true in general.To overcome this difficulty in the CR case, we follow the classical (yet nonK¨ahler) case and introducesome kind of CR Spin c Dirac operator on CR manifolds with S action: e D b,m = ∂ b + ∂ ∗ b + zeroth order termwith modified/ Spin c Kohn Laplacians e (cid:3) + b,m = e D ∗ b,m e D b,m , e (cid:3) − b,m = e D b,m e D ∗ b,m .A word of caution is in order. The above adaptation of the idea of Spin c structure to our CR caseis not altogether straightforward. Locally X is realized as a (portion of a) circle bundle over a smallpiece of complex manifold (via BRT charts), so presumably there could arise a problem of patching upwhen this global Spin c operator is to be formed. See Proposition 4.2 for more.We will show in Theorem 4.7 the homotopy invariance for the index of ∂ b + ∂ ∗ b , and in Corollary 4.8a McKean-Singer formula for the modified Kohn Laplacians: for t > ,(1.93) n X j =0 ( − j dim H jb,m ( X, E ) = Z X (cid:16) Tr e − t e (cid:3) + b,m ( x, x ) − Tr e − t e (cid:3) − b,m ( x, x ) (cid:17) dv X . For u ∈ Ω , ± m ( X ) we can write (on D ) u ( z, θ ) = e − imθ e u ( z ) for some e u ( z ) ∈ Ω , ± ( U, L m ) with D in aBRT trivialization B := ( D, ( z, θ ) , ϕ ) .A fundamental relation that we will show in Proposition 5.1, based on Proposition 4.2, is that(1.94) e − mϕ e (cid:3) ± B,m ( e mϕ e u ) = e imθ e (cid:3) ± b,m ( u ) where e (cid:3) ± B,m = D ∗ B,m D B,m : Ω , ± ( U, L m ) → Ω , ± ( U, L m ) and D B,m : Ω , ± ( U, L m ) → Ω , ∓ ( U, L m ) the (ordinary) Spin c Dirac operator (cf. Definition 4.1) with respect to the Chern connection on L m (induced by h L m ) and the Clifford connection on Λ( T ∗ , U ) (induced by the given Hermitian metric on Λ( T ∗ , U ) ).It is conceivable that X with the CR structure and X/S = M with the complex structure (if defined)are linked in some way (as Theorem 1.2). To say more, the result (1.94) asserts a fundamental factthat not only complex/CR geometrically can the two spaces be linked, but metrically in the sense ofLaplacians they also can. This link is important for our Spin c approach to the CR case to be possible. EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 31
In the remaining let’s give an outline with the CR
Spin c Dirac operator when X is not CR K¨ahler.Although the following ingredients mostly parallel those in the preceding Subsection 1.7.3, the successof this method relies on, among others, the Spin c structure and the associated Clifford connection. Forthat reason and for the sake of clarity, we prefer to put down the precise formulas despite the greatsimilarity in expressions as above.As (1.82), there exists (modified) e A B j , + ( t, z, w ) such that lim t → e A B j , + ( t ) = I in D ′ ( U j , T ∗ , + U j ) , e A ′ B j , + ( t ) u + e A B j , + ( t )( e (cid:3) + B j ,m u ) = 0 , ∀ u ∈ Ω , +0 ( U j ) , ∀ t > , (1.95)and e A B j , + ( t, z, w ) admits an asymptotic expansion as t → + (see (5.19)). Put e H j ( t, x, y ) = χ j ( x ) e − mϕ j ( z ) − imθ e A B j , + ( t, z, w ) e mϕ j ( w )+ imη e χ j ( y ) , e Γ( t ) = N X j =1 e H j ( t ) ◦ Q m . (1.96)Similar to (1.85) and (1.86) in Subsection 1.7.3, one has(1.97) (cid:13)(cid:13)(cid:13) e − t e (cid:3) + b,m ( x, y ) − e Γ( t, x, y ) (cid:13)(cid:13)(cid:13) C ( X × X ) ≤ e − ǫ t , ∀ t ∈ (0 , ǫ ) and(1.98) e Γ( t, x, x ) ∼ (cid:16) p X s =1 e π ( s − p mi (cid:17) ∞ X j =0 t − n + j e α + n − j ( x ) mod O (cid:16) t − n e − ε d ( x,X sing )2 t (cid:17) , ∀ x ∈ X p , with some constants ε , ε , ε > , giving(1.99) e − t e (cid:3) + b,m ( x, x ) ∼ (cid:16) p X s =1 e π ( s − p mi (cid:17) ∞ X j =0 t − n + j e α + n − j ( x ) mod O (cid:16) t − n e − ε d ( x,X sing )2 t (cid:17) on X p . Similar results hold for e − t e (cid:3) − b,m ( x, x ) .The novelty here is analogous to (1.87). By taking supertrace we can improve the estimates in(1.99) (see Theorem 6.4) so that t − n is removed: Tr e − t e (cid:3) + b,m ( x, x ) − Tr e − t e (cid:3) − b,m ( x, x ) ∼ (cid:16) p X s =1 e π ( s − p mi (cid:17) ∞ X j =0 t − n + j (cid:16) Tr e α + n − j ( x ) − Tr e α − n − j ( x ) (cid:17) mod O (cid:16) e − ˆ d ( x,X sing )2 t (cid:17) , (1.100)for x ∈ X p . Hence (1.93) and (1.100) give(1.101) n X j =0 ( − j dim H jb,m ( X, E ) = (cid:16) p X s =1 e π ( s − p mi (cid:17) lim t → + Z X n X ℓ =0 t − ℓ (cid:16) Tr e α + ℓ ( x ) − Tr e α − ℓ ( x ) (cid:17) dv X ( x ) . A key advantage of introducing our CR
Spin c Dirac operator is basically that
Lichnerowicz formulas hold for e (cid:3) + B,m (and e (cid:3) − B,m ). This enables us to apply the rescaling technique (this part of rescaling isessentially the same as in classical cases, cf. [6] and [23]) and to obtain that for each j = 1 , , . . . , N , (cid:16) Tr e A B j , + ( z, z ) − Tr e A B j , − ( z, z ) (cid:17) dv U j ( z )= [Td ( ∇ T , U j , T , U j ) ∧ ch ( ∇ L m , L m )] n ( z ) + O ( t ) , ∀ z ∈ U j . (1.102) EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 32
Rewriting (1.102) in tangential forms, one has n X ℓ =0 t − ℓ (cid:16) Tr e α + ℓ ( x ) − Tr e α − ℓ ( x ) (cid:17) dv X ( x )= 12 π h Td b ( ∇ T , X , T , X ) ∧ e − m dω π ∧ ω i n +1 ( x ) + O ( t ) (1.103)for t > and x ∈ X p .Theorem 1.3, Theorem 1.10 and Corollary 1.13 follows from (1.99), (1.100), (1.101) and (1.103).The layout of this paper is as follows. In Section 2.1 and Section 2.2, we collect some notations, def-initions, terminologies and statements we use throughout. In Section 2.3, we introduce the tangentialde Rham cohomology group, tangential Chern character and tangential Todd class on CR manifoldswith S action. In Section 2.4, we recall a classical result of Baouendi-Rothschild-Treves [4] whichplays an important role in our construction of the heat kernel. We also prove that for a rigid vectorbundle F over X there exist rigid Hermitian metric and rigid connection on F . In Section 3, weestablish a Hodge theory for Kohn Laplacian in the L space of the m -th S Fourier component. InSection 4, we introduce our CR
Spin c Dirac operator e D b,m , modified/ Spin c Kohn Laplacians e (cid:3) ± b,m andprove (1.93). In Section 5, we construct approximate heat kernels for the operators e − t e (cid:3) ± b,m and provethat e − t e (cid:3) ± b,m ( x, y ) admit asymptotic expansions in the sense as (1.97). In Section 6, we prove (1.98),(1.100), (1.103) and finish the proofs of Theorem 1.3, Theorem 1.10 and Corollary 1.13. In Section 7we prove Theorem 1.14. Part I: Preparatory foundations
2. P
RELIMINARIES
Some standard notations.
We use the following notations: N = { , , . . . } , N = N ∪ { } , R is the set of real numbers, R + := { x ∈ R ; x ≥ } . For a multiindex α = ( α , . . . , α n ) ∈ N n we set | α | = α + . . . + α n . For x = ( x , . . . , x n ) we write x α = x α . . . x α n n , ∂ x j = ∂∂x j , ∂ αx = ∂ α x . . . ∂ α n x n = ∂ | α | ∂x α ,D x j = 1 i ∂ x j , D αx = D α x . . . D α n x n , D x = 1 i ∂ x . Let z = ( z , . . . , z n ) , z j = x j − + ix j , j = 1 , . . . , n , be coordinates of C n . We write z α = z α . . . z α n n , z α = z α . . . z α n n ,∂ z j = ∂∂z j = 12 (cid:16) ∂∂x j − − i ∂∂x j (cid:17) , ∂ z j = ∂∂z j = 12 (cid:16) ∂∂x j − + i ∂∂x j (cid:17) ,∂ αz = ∂ α z . . . ∂ α n z n = ∂ | α | ∂z α , ∂ αz = ∂ α z . . . ∂ α n z n = ∂ | α | ∂z α . Let X be a C ∞ orientable paracompact manifold. We denote the tangent and cotangent bundle of X by T X and T ∗ X respectively, and the complexified tangent and cotangent bundle by C T X and C T ∗ X .We write h · , · i to denote the pointwise pairing between T ∗ X and T X and extend h · , · i bilinearly to C T ∗ X × C T X .Let E , F be C ∞ vector bundles over X . We write F ⊠ E ∗ for the vector bundle over X × X withfiber over ( x, y ) ∈ X × X consisting of linear maps from E y to F x .Let Y ⊂ X be an open subset. The spaces of smooth sections and distribution sections of E over Y will be denoted by C ∞ ( Y, E ) and D ′ ( Y, E ) respectively. Let E ′ ( Y, E ) be the subspace of D ′ ( Y, E ) EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 33 whose elements are of compact support in Y . For m ∈ R , we let H m ( Y, E ) denote the Sobolev spaceof order m for sections of E over Y . Put H m loc ( Y, E ) = (cid:8) u ∈ D ′ ( Y, E ); ϕu ∈ H m ( Y, E ) , ϕ ∈ C ∞ ( Y ) (cid:9) ,H m comp ( Y, E ) = H m loc ( Y, E ) ∩ E ′ ( Y, E ) . Set up and terminology.
Let ( X, T , X ) be a compact CR manifold of dimension n + 1 , n ≥ ,where T , X is a CR structure of X . That is T , X is a subbundle of rank n of the complexifiedtangent bundle C T X , satisfying T , X ∩ T , X = { } , where T , X = T , X , and [ V , V ] ⊂ V , V = C ∞ ( X, T , X ) .We assume that X admits an S action: S × X → X . We write e − iθ to denote the S action.Let T ∈ C ∞ ( X, T X ) be the global real vector field induced by the S action given by ( T u )( x ) = ∂∂θ (cid:0) u ( e − iθ ◦ x ) (cid:1) | θ =0 for u ∈ C ∞ ( X ) . Definition 2.1.
We say that the S action e − iθ is CR if [ T, C ∞ ( X, T , X )] ⊂ C ∞ ( X, T , X ) and the S action is transversal if for each x ∈ X , C T ( x ) ⊕ T , x X ⊕ T , x X = C T x X . Moreover, we say that the S action is locally free if T = 0 everywhere.We assume throughout that ( X, T , X ) is a compact CR manifold with a transversal CR locally free S action e − iθ with T the global vector field induced by the S action. Let ω ∈ C ∞ ( X, T ∗ X ) be theglobal real one form determined by h ω , u i = 0 for all u ∈ T , X ⊕ T , X , and h ω , T i = 1 . Definition 2.2.
For p ∈ X , the Levi form L p is the Hermitian quadratic form on T , p X given by L p ( U, V ) = − i h dω ( p ) , U ∧ V i , U, V ∈ T , p X .If the Levi form L p is semi-positive definite (resp. positive definite), we say that X is weakly pseu-doconvex (resp. strongly pseudoconvex) at p . If the Levi form is semi-positive definite (resp. positivedefinite) at every point of X , we say that X is weakly pseudoconvex (resp. strongly pseudoconvex).Denote by T ∗ , X and T ∗ , X the dual bundles of T , X and T , X respectively. Define the vectorbundle of ( p, q ) forms by T ∗ p,q X = Λ p ( T ∗ , X ) ∧ Λ q ( T ∗ , X ) .Let D ⊂ X be an open subset and E be a complex vector bundle over D . Denote by Ω p,q ( D, E ) (resp. Ω p,q ( D ) ) the space of smooth sections of T ∗ p,q X ⊗ E (resp. T ∗ p,q X )) over D and by Ω p,q ( D, E ) (resp. Ω p,q ( D ) ) those elements of compact support in D .Put T ∗ , • X := ⊕ j ∈{ , ,...,n } T ∗ ,j X,T ∗ , + X := ⊕ j ∈{ , ,...,n } , j is even T ∗ ,j X,T ∗ , − X := ⊕ j ∈{ , ,...,n } , j is odd T ∗ ,j X. Put Ω , • ( X, E ) , Ω , + ( X, E ) and Ω , − ( X, E ) in a similar way as above.Fix θ ∈ ] − π, π [ . Let ( e − iθ ) ∗ : Λ r ( C T ∗ X ) → Λ r ( C T ∗ X ) be the pull-back map, ( e − iθ ) ∗ : T ∗ p,qe − iθ ◦ x X → T ∗ p,qx X . Define for u ∈ Ω p,q ( X ) (2.1) T u := ∂∂θ (cid:0) ( e − iθ ) ∗ u (cid:1) | θ =0 ∈ Ω p,q ( X ) . (See also (2.13).)We shall write u ( e − iθ ◦ x ) := ( e − iθ ) ∗ u ( x ) for u ∈ Ω p,q ( X ) . Clearly(2.2) u ( x ) = X m ∈ Z π Z π − π u ( e − iθ ◦ x ) e imθ dθ. Let ∂ b : Ω ,q ( X ) → Ω ,q +1 ( X ) be the tangential Cauchy-Riemann operator. From the CR propertyof the S action it follows that (see also (2.14)) T ∂ b = ∂ b T on Ω ,q ( X ) . EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 34
Naturally associated with the S action are the so-called rigid objects. See also [4] for a similar useof this term (cf. Definition II.2 of loc.cit. ). Definition 2.3.
Let D ⊂ X be an open set and u ∈ C ∞ ( D ) . We say that u is rigid if T u = 0 , u isCauchy-Riemann (CR for short) if ∂ b u = 0 and u is a rigid CR function if ∂ b u = 0 and T u = 0 . Definition 2.4.
Let F be a complex vector bundle of rank r over X . We say that F is rigid (resp.CR) if X can be covered by open subsets U j with trivializing frames { f j , f j , . . . , f rj } such that thecorresponding transition functions are rigid (resp. CR) (in the sense of the preceding definition). Inthis case the frames { f j , f j , . . . , f rj } are called rigid frames (resp. CR frames).Let F be a rigid complex vector bundle over X in the sense of Definition 2.4. Definition 2.5.
Let h · | · i F be a Hermitian metric on F . We say that h · | · i F is a rigid Hermitian metric if for every rigid local frames { f , . . . , f r } of F , we have T h f j | f k i F = 0 , for j, k = 1 , , . . . , r .The condition of being rigid is not a severe restriction as far as the S action is concerned. SeeTheorems 2.11 and 2.12 which we shall prove within the framework of BRT trivializations in the nextsection.Henceforth let E be a rigid CR vector bundle over X . Write ∂ b : Ω ,q ( X, E ) → Ω ,q +1 ( X, E ) forthe tangential Cauchy-Riemann operator. Since E is rigid, we can define T u for u ∈ Ω ,q ( X, E ) (cf.Theorem 2.11) and have(2.3) T ∂ b = ∂ b T on Ω ,q ( X, E ) . For m ∈ Z , let(2.4) Ω ,qm ( X, E ) := (cid:8) u ∈ Ω ,q ( X, E ); T u = − imu (cid:9) and put Ω , • m ( X, E ) , Ω , + m ( X, E ) and Ω , − m ( X, E ) in a similar way as above.Put ∂ b,m := ∂ b : Ω ,qm ( X, E ) → Ω ,q +1 m ( X, E ) with a ∂ b,m -complex: ∂ b,m : · · · → Ω ,q − m ( X, E ) → Ω ,qm ( X, E ) → Ω ,q +1 m ( X, E ) → · · · . Define H qb,m ( X, E ) := Ker ∂ b,m : Ω ,qm ( X, E ) → Ω ,q +1 m ( X, E )Im ∂ b,m : Ω ,q − m ( X, E ) → Ω ,qm ( X, E ) . It is instructive to think of H qb,m ( X, E ) as the m -th S Fourier component of the q -th ∂ b Kohn-Rossicohomology group.We will prove in Theorem 3.7 that dim H qb,m ( X, E ) < ∞ , for m ∈ Z and q = 0 , , , . . . , n .We take a rigid Hermitian metric h · | · i E on E (in the sense of Definition 2.5), and a rigid Hermitianmetric h · | · i on C T X such that(2.5) T ⊥ ( T , X ⊕ T , X ) , h T | T i = 1 (and T , X ⊥ T , X ). (This is always possible; see Theorem 2.11 and Theorem 9.2 in [41].)The Hermitian metric h · | · i on C T X induces by duality a Hermitian metric on C T ∗ X and on thebundles of (0 , q ) forms T ∗ ,q X ( q = 0 , · · · , n ), to be denoted by h · | · i too. A Hermitian metricdenoted by h · | · i E on T ∗ , • X ⊗ E is induced by those on T ∗ , • X and E . Let the lnear map A ( x, y ) ∈ ( T ∗ , • X ⊗ E ) ⊠ ( T ∗ , • X ⊗ E ) ∗ | ( x,y ) . We write | A ( x, y ) | to denote the natural matrix norm of A ( x, y ) induced by h · | · i E .We denote by dv X = dv X ( x ) the induced volume form, and form the global L inner products ( · | · ) E and ( · | · ) on Ω , • ( X, E ) and Ω , • ( X ) respectively, with L -completion L ( X, T ∗ ,q X ⊗ E ) and L ( X, T ∗ ,q X ) . Similar notation applies to L m ( X, T ∗ ,q X ⊗ E ) and L m ( X, T ∗ ,q X ) (the completionsof Ω ,qm ( X, E ) and Ω ,qm ( X ) with respect to ( · | · ) E and ( · | · ) ).Put L ( X, T ∗ , • X ⊗ E ) , L , + ( X, E ) and L , − ( X, E ) in a similar way as above, and L m ( X, T ∗ , • X ⊗ E ) , L , + m ( X, E ) and L , − m ( X, E ) too. EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 35
Tangential de Rham cohomology group, Tangential Chern character and Tangential Toddclass.
In this section it is convenient to put Ω r ( X ) = { u ∈ ⊕ p + q = r Ω p,q ( X ); T u = 0 } for r = 0 , , , . . . , n (without the danger of confusion with Ω p,q in the preceding section) and set Ω • ( X ) = ⊕ nr =0 Ω r ( X ) .Since T d = dT (see (2.3)), we have d -complex: d : · · · → Ω r − ( X ) → Ω r ( X ) → Ω r +10 ( X ) → · · · Define the r -th tangential de Rham cohomology group: H rb, ( X ) := Ker d : Ω r ( X ) → Ω r +10 ( X )Im d : Ω r − ( X ) → Ω r ( X ) . Put H • b, ( X ) = ⊕ nr =0 H rb, ( X ) .Let a complex vector bundle F over X of rank r be rigid as in Definition 2.4. We will show in Theo-rem 2.12 that there exists a connection ∇ on F such that for any rigid local frame f = ( f , f , . . . , f r ) of F on an open set D ⊂ X , the connection matrix θ ( ∇ , f ) = ( θ j,k ) rj,k =1 satisfies θ j,k ∈ Ω ( D ) , for j, k = 1 , . . . , r . We call ∇ as such a rigid connection on F . Let Θ( ∇ , F ) ∈ C ∞ ( X, Λ ( C T ∗ X ) ⊗ End ( F )) be the associated tangential curvature .Let h ( z ) = P ∞ j =0 a j z j be a real power series on z ∈ C . Set H (Θ( ∇ , F )) = Tr (cid:16) h (cid:0) i π Θ( ∇ , F ) (cid:1)(cid:17) . It is clear that H (Θ( ∇ , F )) ∈ Ω ∗ ( X ) .The following is well-known (see Theorem B.5.1 in Ma-Marinescu [49]). Theorem 2.6. H (Θ( ∇ , F )) is a closed differential form. That the tangential de Rham cohomology class [ H (Θ( ∇ , F ))] ∈ H • b, ( X ) does not depend on the choice of rigid connections ∇ is given by Theorem 2.7.
Let ∇ ′ be another rigid connection on F . Then, H (Θ( ∇ , F )) − H (Θ( ∇ ′ , F )) = dA , forsome A ∈ Ω ∗ ( X ) .Proof. The idea of the proof is standard. For each t ∈ [0 , , put ∇ t = (1 − t ) ∇ + t ∇ ′ which is a rigidconnection on F . Set Q t = i π Tr (cid:16) ∂ ∇ t ∂t h ′ (cid:0) i π Θ( ∇ t , F ) (cid:1)(cid:17) . Since ∇ t is rigid, it is easily seen that(2.6) Q t ∈ Ω • ( X ) . It is well-known that (see Remark B.5.2 in Ma-Marinescu [49])(2.7) H (Θ( ∇ , F )) − H (Θ( ∇ ′ , F )) = d Z Q t dt. From (2.6) and (2.7), the theorem follows. (cid:3)
For h ( z ) = e z put(2.8) ch b ( ∇ , F ) := H (Θ( ∇ , F )) ∈ Ω • ( X ) , and for h ( z ) = log( z − e − z ) set(2.9) Td b ( ∇ , F ) := e H (Θ( ∇ ,F )) ∈ Ω • ( X ) . We can now introduce tangential Todd class and tangential Chern character.
EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 36
Definition 2.8.
The tangential Chern character of F is given by ch b ( F ) := [ch b ( ∇ , F )] ∈ H • b, ( X ) and the tangential Todd class of F is given by Td b ( F ) = [Td b ( ∇ , F )] ∈ H • b, ( X ) . Baouendi-Rothschild-Treves [4] proved that T , X is a rigid complex vector bundle over X (cf. thefirst part of Theorem 2.11 below). The tangential Todd class of T , X and tangential Chern characterof T , X are thus well defined.The tangential Chern classes can be defined similarly. Put det( i Θ( ∇ ,F )2 π t + I ) = r P j =0 ˆ c j ( ∇ , F ) t j .Thus ˆ c j ( ∇ , F ) ∈ Ω j ( D ) . By the matrix identity det A = e Tr (log A ) and taking h ( z ) = log(1 + z ) ,one sees ˆ c j ( ∇ , F ) ( j = 0 , , . . . , r ) is a closed differential form on X and its tangential de Rhamcohomology class [ˆ c j ( ∇ , F )] ∈ H jb, ( X ) is independent of the choice of rigid connections ∇ . Put ˆ c j ( F ) = [ˆ c j ( ∇ , F )] ∈ H jb, ( X ) . We call ˆ c j ( F ) the j -th tangential Chern class of F , and ˆ c ( F ) =1 + r P j =1 ˆ c j ( F ) ∈ H • b, ( X ) the tangential total Chern class of F .2.4. BRT trivializations and rigid geometric objects.
In this paper, much of our strategy is heavilybased on the following result thanks to Baouendi-Rothschild-Treves [4, Proposition I.2]. Note in thefollowing, Z j corresponds to L j in their proposition. Some geometrical significance related to a certaincircle bundle structure will be discussed in the proof of Proposition 4.2. Theorem 2.9.
For every point x ∈ X there exist local coordinates x = ( x , · · · , x n +1 ) = ( z, θ ) =( z , · · · , z n , θ ) , z j = x j − + ix j , j = 1 , · · · , n, x n +1 = θ , defined in some small neighborhood D = { ( z, θ ) : | z | < δ, − ε < θ < ε } of x , δ > , < ε < π , such that ( z ( x ) , θ ( x )) = (0 , and T = ∂∂θZ j = ∂∂z j − i ∂ϕ∂z j ( z ) ∂∂θ , j = 1 , · · · , n (2.10) where Z j ( x ) , j = 1 , · · · , n , form a basis of T , x X for each x ∈ D and ϕ ( z ) ∈ C ∞ ( D, R ) is independentof θ . We summarize these data by the notation ( D, ( z, θ ) , ϕ ) .Furthermore, let ( D, ( z, θ ) , ϕ ) and ( e D, ( w, η ) , e ϕ ) be two such data on D . Then the coordinate transfor-mation between them (on D ∩ e D ) can be given such that if w = ( w , . . . , w n ) = H ( z ) = ( H ( z ) , . . . , H n ( z )) then H j ( z ) ∈ C ∞ ( | z | < δ ) , ∂H j ( z ) = 0 , ∀ jη = θ + arg g ( z ) (mod 2 π ) where arg g ( z ) = Im log g ( z ) e ϕ ( H ( z ) , H ( z )) = ϕ ( z, z ) + log | g ( z ) | (2.11) for some nowhere vanishing holomorphic function g ( z ) on | z | < δ .Remark . The relation between ˜ ϕ and ϕ in (2.11) is a corrected version of a similar formula in [4,the line below (I.31)]. See the proof of Proposition 4.2 for a derivation.There exist examples that H is not necessarily one to one. Nevertheless, it can be shown that aftershrinking D and e D properly, it is one to one, hence a biholomorphism.We call the above triple ( D, ( z, θ ) , ϕ ) a BRT trivialization . Note for ( z, θ ) ∈ D and − π < α < π , e − iα ◦ ( z, θ ) = ( z, θ + α ) if { e − itα ◦ ( z, θ ) } ≤ t ≤ ⊂ D . EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 37
By using BRT trivializations some operations simplify, as follows. Under the BRT triple ( D, ( z, θ ) , ϕ ) it is clear that { dz j ∧ · · · ∧ dz j q , ≤ j < · · · < j q ≤ n } is a basis for T ∗ ,qx X for every x ∈ D . For u ∈ Ω ,q ( X ) , on D we write(2.12) u = X j < ··· Suppose F is a complex vector bundle over X (not necessarily a CR bundle) and admitsan S action compatible with that on X . Then F is actually a rigid vector bundle (in the sense ofDefinition 2.4). Moreover there is a rigid Hermitian metric h · | · i F on F . Conversely if F is a rigid vectorbundle, then F admits a compatible S action.Proof. We first work on the existence of a rigid Hermitian metric (assuming F is rigid). Fix p ∈ X andlet ( D, ( z, θ ) , ϕ ) be a BRT trivialization around p such that ( z ( p ) , θ ( p )) = (0 , , ( z, θ ) ∈ { z ∈ C n − : | z | < δ } × { θ ∈ R : | θ | < δ } for some δ > . Put A := { λ ∈ [ − π, π ] : there is a local rigid trivializing frame (l.r.t. frame for short ) defined on (cid:8) e − iθ ◦ ( z, | z | < ε, θ ∈ [ − π, λ + ε ) (cid:9) for some < ε < δ } . Clearly A is a non-empty open set in [ − π, π ] . We claim A = [ − π, π ] . (Remark that the l.r.t. frameabove is closely related to the canonical basis in [4, Definition I.3 without (I.29a)] when E is T , X .)It suffices to prove A is closed. Let λ be a limit point of A . For some small ε > , there is al.r.t. frame ˆ f = ( ˆ f , . . . , ˆ f r ) defined on (cid:8) e − iθ ◦ ( z, | z | < ε , λ − ε < θ < λ + ε (cid:9) . By assumption λ ∈ A there exists a l.r.t. frame e f = ( e f , . . . , e f r ) defined around { e − iθ ◦ ( z, } in which | z | < ε , θ ∈ [ − π, λ − ε ) for some ε > . Now e f = g ˆ f on n e − iθ ◦ ( z, | z | < ε , θ ∈ ( λ − ε , λ − ε o , ε = min { ε , ε } for some rigid r × r matrix g .We now patch up the frames. Put , for θ ∈ [ − π, λ − ε ) , f = e f (on { e − iθ ◦ ( z, } ) and for θ ∈ [ λ − ε , λ + ε ) , f = g ˆ f because g is independent of θ . By e f = g ˆ f on the overlapping, f iswell-defined as a l.r.t. frame on n e − iθ ◦ ( z, | z | < ε , θ ∈ [ − π, λ + ε ) o . extending θ = λ . Thus A is closed as desired.By the discussion above we can actually find local rigid trivializations W , . . . , W N such that X = S Nj =1 W j and each W j ⊃ S − π ≤ θ ≤ π e − iθ W j (i.e. W j is S invariant). Take any Hermitian metric h · , · i F on F . Let h · | · i F be the Hermitian metric on F defined as follows. For each j = 1 , , . . . , N , let EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 38 h j , . . . , h rj be local rigid trivializing frames on W j . Put h h sj ( x ) | h tj ( x ) i F = π R π − π h h sj ( e − iu ◦ x ) , h tj ( e − iu ◦ x ) i F du , s, t = 1 , , . . . , r . One sees that h · | · i F is well-defined as a rigid Hermitian metric on F .By examining the above reasoning we have also proved that if F is rigid, then it admits a natural S action (by declaring the l.r.t. frames as S invariant frames) compatible with that on X .For the reverse direction if F admits a compatible S action, by using BRT trivializations one canconstruct S invariant local frames. These invariant local frames can be easily verified to be localrigid frames, or equivalently the transition functions between them are annihilated by T due to the S invariant property. Hence F is rigid by definition. (cid:3) We shall now prove Theorem 2.12. Assume the complex vector bundle F is rigid (on X ). There exists a rigid connectionon F . And if F is equipped with a rigid Hermitian metric, there exists a rigid connection compatiblewith this Hermitian metric. Suppose F is furthermore CR (and rigid) equipped with a rigid Hermitianmetric h . Then there exists a unique rigid connection (see the second paragraph of Subsection 2.3) ∇ F compatible with h such that ∇ F induces a Chern connection on U j (of any given BRT chart), and that the S invariant sections are its parallel sections along S orbits of X .Proof. Let ∇ be a connection on F . For any g ∈ S considering g ∗ ∇ on g ∗ F which is F by usingTheorem 2.11 and summing over g (in analogy with the construction of a rigid Hermitian metricabove), one obtains a rigid connection on F . Suppose F has a rigid Hermitian metric h · | · i F and aconnection ∇ compatible with h · | · i F . One readily sees that the rigid connection resulting from thepreceding procedure of summation, is still compatible with h · | · i F . For the last statement, note that i)given a rigid CR bundle F with a rigid Hermitian metric h and any BRT chart D j = U j × ] − ε, ε [ , ( F, h ) can descend to U j as a holomorphic vector bundle with the inherited metric, and ii) for a holomorphicvector bundle with a Hermitian metric, the Chern connection is canonically defined. Combining i)with ii) and using the S invariant local frames (cf. proof of Theorem 2.11), one can construct acanonical connection ∇ Fj on U j . Then by using (2.10), (2.11) and the canonical property of the Chernconnection, one sees that these ∇ Fj patch up to form a global connection ∇ F on X , satisfying theproperty as stated in the proposition. (cid:3) 3. A H ODGE THEORY FOR (cid:3) ( q ) b,m A Hodge decomposition theorem for ∂ b on pseudoconvex CR manifolds has been well developed.See [15, Section 9.4] for a nice presentation in some respects; see also [59]. Our goal of this sectionis to develop an analogous theory for (cid:3) ( q ) b,m on CR manifolds with transversal CR locally free S action(irrespective of pseudoconvexity). Much of what follows appears to parallel the corresponding part ofHodge theory in complex geometry.Besides the relevance to the index theorem on CR manifolds, the present theory has an applicationto our proof of homotopy invariance of index (Theorem 4.7).As before, X is a compact CR manifold with a transversal CR locally free S action. Let ∂ ∗ b :Ω ,q +1 ( X, E ) → Ω ,q ( X, E ) ( q = 0 , , , . . . , n ) be the formal adjoint of ∂ b with respect to ( · | · ) E . Put (cid:3) ( q ) b := ∂ b ∂ ∗ b + ∂ ∗ b ∂ b : Ω ,q ( X, E ) → Ω ,q ( X, E ) . T is the vector field on X induced by the S action, T ∂ b = ∂ b T and ∂ b,m := ∂ b | Ω ,qm : Ω ,qm ( X, E ) → Ω ,q +1 m ( X, E ) on eigenspaces of the S acton ( ∀ m ∈ Z ) .Recall h · | · i E is rigid. One sees T ∂ ∗ b = ∂ ∗ b T so that ∂ ∗ b | Ω ,q +1 m : Ω ,q +1 m ( X, E ) → Ω ,qm ( X, E ) is thesame as the formal adjoint ∂ ∗ b,m of ∂ b,m . Form (cid:3) ( q ) b,m = ∂ b,m ∂ ∗ b,m + ∂ ∗ b,m ∂ b,m : Ω ,qm ( X, E ) → Ω ,qm ( X, E ) .We have (cid:3) ( q ) b,m = (cid:3) ( q ) b | Ω ,qm ( X,E ) .On a general compact CR manifold, there is a fundamental result that follows from Kohn’s L estimates. (See [15, Theorem 8.4.2]). Adapting it to our present situation, we can state the result asfollows. EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 39 Theorem 3.1. For every s ∈ N , there is a constant C s > such that k u k s +1 ≤ C s (cid:16) (cid:13)(cid:13)(cid:13) (cid:3) ( q ) b u (cid:13)(cid:13)(cid:13) s + k T u k s + k u k s (cid:17) , ∀ u ∈ Ω ,q ( X, E ) , where k·k s denotes the usual Sobolev norm of order s on X . Theorem 3.1 restricted to Ω ,qm ( X, E ) yields Corollary 3.2. Fix m ∈ Z . For every s ∈ N , there is a constant C s > such that k u k s +1 ≤ C s (cid:16) (cid:13)(cid:13)(cid:13) (cid:3) ( q ) b,m u (cid:13)(cid:13)(cid:13) s + k u k s (cid:17) , ∀ u ∈ Ω ,qm ( X, E ) . This suggests that a good regularity theory might exist on our X . Observe that (cid:3) ( q ) b − T is elliptic on X while (cid:3) ( q ) b is not, which on Ω ,qm ( X, E ) is (cid:3) ( q ) b,m + m . In fact, without using the above theoremsall of the following results are essentially proven by standard results in elliptic theory.Write Dom (cid:3) ( q ) b,m := { u ∈ L m ( X, T ∗ ,q X ⊗ E ); (cid:3) ( q ) b,m u ∈ L m ( X, T ∗ ,q X ⊗ E ) } where (cid:3) ( q ) b,m u is definedin the sense of distribution. (cid:3) ( q ) b,m is extended by(3.1) (cid:3) ( q ) b,m : Dom (cid:3) ( q ) b,m ( ⊂ L m ( X, T ∗ ,q X ⊗ E )) → L m ( X, T ∗ ,q X ⊗ E ) . Lemma 3.3. We have Dom (cid:3) ( q ) b,m = L m ( X, T ∗ ,q X ⊗ E ) T H ( X, T ∗ ,q X ⊗ E ) .Proof. For the inclusion put v = (cid:3) ( q ) b,m u ∈ L m ( X, T ∗ ,q X ⊗ E ) . Then ( (cid:3) ( q ) b,m − T ) u = v + m u ∈ L m ( X, T ∗ ,q X ⊗ E ) . Since ( (cid:3) ( q ) b − T ) is elliptic, we conclude u ∈ H ( X, T ∗ ,q X ⊗ E ) . The reverseinclusion is clear. (cid:3) Lemma 3.4. (cid:3) ( q ) b,m : Dom (cid:3) ( q ) b,m ( ⊂ L m ( X, T ∗ ,q X ⊗ E )) → L m ( X, T ∗ ,q X ⊗ E ) is self-adjoint.Proof. Since the similar extension of (cid:3) ( q ) b on L ( X, T ∗ ,q X ⊗ E ) is self-adjoint and its restriction to (aninvariant subspace) L m ( X, T ∗ ,q X ⊗ E ) gives (cid:3) ( q ) b,m , (cid:3) ( q ) b,m is also self-adjoint. (cid:3) Let Spec (cid:3) ( q ) b,m ⊂ [0 , ∞ [ denote the spectrum of (cid:3) ( q ) b,m (Davies [19]). Proposition 3.5. Spec (cid:3) ( q ) b,m is a discrete subset of [0 , ∞ [ . For any ν ∈ Spec (cid:3) ( q ) b,m , ν is an eigenvalue of (cid:3) ( q ) b,m and the eigenspace E qm,ν ( X, E ) := (cid:8) u ∈ Dom (cid:3) ( q ) b,m ; (cid:3) ( q ) b,m u = νu (cid:9) is finite dimensional with E qm,ν ( X, E ) ⊂ Ω ,qm ( X, E ) .Proof. (cid:3) ( q ) b − T ≡ ∆ is a second order elliptic operator. By standard elliptic theory, ∆ and hence ∆ + m , satisfy the statement of the proposition on the (invariant) subspace L m ( X, T ∗ ,q X ⊗ E ) ⊃ Ω ,qm ( X, E ) . On it T acts as − m , the proposition follows for (cid:3) ( q ) b,m which is ∆ + m on L m ( X, T ∗ ,q X ⊗ E ) . (cid:3) A role analogous to the Green’s operator in the ordinary Hodge theory is given as follows. Let N ( q ) m : L m ( X, T ∗ ,q X ⊗ E ) → Dom (cid:3) ( q ) b,m be the partial inverse of (cid:3) ( q ) b,m and let Π ( q ) m : L m ( X, T ∗ ,q X ⊗ E ) → Ker (cid:3) ( q ) b,m be the orthogonal projection. We have (cid:3) ( q ) b,m N ( q ) m + Π ( q ) m = I on L m ( X, T ∗ ,q X ⊗ E ) ,N ( q ) m (cid:3) ( q ) b,m + Π ( q ) m = I on Dom (cid:3) ( q ) b,m . (3.2) EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 40 Lemma 3.6. We have N ( q ) m : Ω ,qm ( X, E ) → Ω ,qm ( X, E ) .Proof. A slight variant of the standard argument applies as (cid:3) ( q ) b is almost elliptic. Let u ∈ Ω ,qm ( X, E ) and put N ( q ) m u = v ∈ L m ( X, T ∗ ,q X ⊗ E ) . By (3.2), ( I − Π ( q ) m ) u = (cid:3) ( q ) b,m v , giving(3.3) ( (cid:3) ( q ) b,m − T ) v = ( I − Π ( q ) m ) u + m v. By Proposition 3.5, Ker (cid:3) ( q ) b,m consists of smooth sections, so Π ( q ) m u is smooth and(3.4) ( I − Π ( q ) m ) u ∈ Ω ,qm ( X, E ) . By combining (3.3) and (3.4) and noting (cid:3) ( q ) b − T is elliptic, the standard technique in elliptic regu-larity applies to give v ∈ Ω ,qm ( X, E ) . (cid:3) The following is a version of “harmonic realization” of cohomology. Theorem 3.7. For every q ∈ { , , , . . . , n } and every m ∈ Z , we have (3.5) Ker (cid:3) ( q ) b,m = E qm, ( X, E ) ∼ = H qb,m ( X, E ) . As a consequence dim H qb,m ( X, E ) < ∞ by Proposition 3.5.Proof. The argument is mostly standard (although (cid:3) ( q ) b is not elliptic). Consider the map τ qm : Ker ∂ b,m \ Ω ,qm ( X, E ) → Ker (cid:3) ( q ) b,m ,u → Π ( q ) m u. (3.6)Clearly τ qm is surjective. Put M qm := n ∂ b,m u ; u ∈ Ω ,q − m ( X, E ) o . The theorem follows if one shows(3.7) Ker τ qm = M qm . It is easily seen M qm ⊂ Ker τ qm since M qm ⊥ Ker (cid:3) ( q ) b,m . For the reverse let u ∈ Ker τ qm so Π ( q ) m u = 0 . From(3.2) we have u = (cid:3) ( q ) b,m N ( q ) m u + Π ( q ) m u = ( ∂ b ∂ ∗ b + ∂ ∗ b ∂ b ) N ( q ) m u. (3.8)We claim that(3.9) ∂ ∗ b ∂ b N ( q ) m u = 0 . One sees, by using ∂ b = 0 for the first equality below, ( ∂ ∗ b ∂ b N ( q ) m u | ∂ ∗ b ∂ b N ( q ) m u ) E = ( ∂ ∗ b ∂ b (cid:3) ( q ) m N ( q ) m u | N ( q ) m u ) E = ( ∂ ∗ b ∂ b ( I − Π ( q ) m ) u | N ( q ) m u ) E = ( ∂ ∗ b ∂ b u | N ( q ) m u ) E (3.10)which is zero because u ∈ Ker ∂ b,m by (3.6), giving the claim (3.9). By (3.8) and (3.9),(3.11) u = ∂ b ∂ ∗ b N ( q ) m u, with ∂ ∗ b N ( q ) m u ∈ Ω ,q − m ( X, E ) by Lemma 3.6. By (3.11), u ∈ M qm , yielding the desired inclusion Ker τ qm ⊂ M qm . Hence (3.7). (cid:3) EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 41 Let D b,m := ∂ b + ∂ ∗ b : Ω , + m ( X, E ) → Ω , − m ( X, E ) with extension D b,m D b,m : Dom D b,m ( ⊂ L , + m ( X, E )) → L , − m ( X, E ) , Dom D b,m = n u ∈ L , + m ( X, E ); distribution D b,m u ∈ L , + m ( X, E ) o . The Hilbert space adjoint of D b,m with respect to ( · | · ) E is given by D ∗ b,m : Dom D ∗ b,m ( ⊂ L , − m ( X, E )) → L , + m ( X, E ) .Combining Proposition 3.5 and Theorem 3.7, one can verify (as in standard Hodge theory) Theorem 3.8. In the notation above Ker D b,m = M q ∈{ , ,...,n } q even Ker (cid:3) ( q ) b,m ( ⊂ Ω , + m ( X, E )) , Ker D ∗ b,m = M q ∈{ , ,...,n } q odd Ker (cid:3) ( q ) b,m ( ⊂ Ω , − m ( X, E )) . (3.12) Put ind D b,m := dim Ker D b,m − dim Ker D ∗ b,m . Hence, together with Theorem 3.7, (3.13) n X j =0 ( − j dim H jb,m ( X, E ) = ind D b,m . 4. M ODIFIED K OHN L APLACIAN ( Spin c K OHN L APLACIAN )We are prepared by Theorem 3.8 above to see that to calculate P nj =0 ( − j dim H jb,m ( X, E ) is thesame as to calculate the index ind D b,m . To do so effectively we need to modify the Dirac type oper-ator D b,m hence the standard Kohn Laplacian because the modified versions e D b,m , e (cid:3) b,m will have amanageable heat kernel that suits our purpose better for the CR non-K¨ahler case (cf. Remark 4.9).Lastly we shall give an argument for the homotopy invariance, and obtain ind D b,m = ind e D b,m .The main idea here is borrowed from that of classical cases. But as the CR manifold X is notassumed to be a (orbifold) circle bundle globally, there could arise the problem of patching (fromlocal constructions to the global one). Part of the technicality in the beginning of this section lies in acareful treatment in this regard.We recall some basics of Clifford connection and Spin c Dirac operator. For more details we refer toChapter 1 in [49] and [23].Let B := ( D, ( z, θ ) , ϕ ) be a BRT trivialization with D = U × ] − ε, ε [ where ε > and U is an openset of C n . Using ϕ in B , we let h · , · i be the Hermitian metric on C T U induced by that on D (4.1) h ∂∂z j , ∂∂z k i = h ∂∂z j − i ∂ϕ∂z j ( z ) ∂∂θ | ∂∂z k − i ∂ϕ∂z k ( z ) ∂∂θ i , j, k = 1 , , . . . , n (cf. Theorem 2.9). By (2.10) and Theorem 2.9, the above metric is actually intrinsically defined.The h · , · i induces Hermitian metrics on T ∗ ,q U still denoted by h · , · i and a Riemannian metric g T U on T U .For any v ∈ T U with decomposition v = v (1 , + v (0 , ∈ T , U ⊕ T , U , let v (1 , , ∗ ∈ T ∗ , U be themetric dual of v (1 , with respect to h · , · i . That is, v (1 , , ∗ ( u ) = h v (1 , , u i for all u ∈ T , U .The Clifford action v on Λ( T ∗ , U ) := ⊕ nq =0 T ∗ ,q U is defined by(4.2) c ( v )( · ) = √ v (1 , , ∗ ∧ ( · ) − i v (0 , ( · )) (where ∧ and i denote the exterior and interior product respectively).Let { w j } nj =1 be a local orthonormal frame of T , U with respect to h · , · i with dual frame (cid:8) w j (cid:9) nj =1 .Write(4.3) e j − = √ ( w j + w j ) and e j = i √ ( w j − w j ) , j = 1 , , . . . , n, EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 42 for an orthonormal frame of T U . Let ∇ T U be the Levi-Civita connection on T U (with respect to g T U ),and ∇ det be the Chern connection on the determinant line bundle det ( T , U ) (with h · , · i ), withconnection forms Γ T U and Γ det associated to the frames { e j } nj =1 and w ∧ · · · ∧ w n . That is, ∇ T Ue j e ℓ = Γ T U ( e j ) e ℓ , j, ℓ = 1 , , . . . , n, ∇ det ( w ∧ · · · ∧ w n ) = Γ det w ∧ · · · ∧ w n . (4.4)The Clifford connection ∇ Cl on Λ( T ∗ , U ) is defined for the frame (cid:8) w j ∧ · · · ∧ w j q ; 1 ≤ j < · · · < j q ≤ n (cid:9) by the local formula(4.5) ∇ Cl = d + 14 n X j,ℓ =1 h Γ T U e j , e ℓ i c ( e j ) c ( e ℓ ) + 12 Γ det . In general a Levi-Civita connection ∇ cannot be compatible with the complex structure unless acertain extra condition is imposed such as K¨ahler condition on the metric. Or else one takes theorthogonal projection P T , X ∇ to produce a connection on T , X . One key point above is that theClifford connection ∇ Cl (regardless of K¨ahler condition nor orthogonal projection) defines a Her-mitian connection (connection compatible with the underlying Hermitian metric) on Λ( T ∗ , U ) (seeProposition 1.3.1 in [49]).Let’s be back to the CR case. In the same notation as before, Ω ,q ( U, E ) denotes the space of (0 , q ) forms on U with values in E , Ω , + ( U, E ) the even part and Ω , − ( U, E ) the odd part of Ω , ∗ ( U, E ) etc.Assume X is equipped with a CR bundle E which is rigid. Being rigid E can descend as a holomor-phic vector bundle over U . We may assume that E is (holomorphically) trivial on U (possibly aftershrinking U ). A rigid Hermitian (fiber) metric h · | · i E descends to a Hermitian (fiber) metric h · | · i E on E over U . Let ∇ E be the Chern connection on E associated with h · | · i E (over U ).We still denote by ∇ Cl the connection on Λ( T ∗ , U ) ⊗ E induced by ∇ Cl and ∇ E . Definition 4.1. The Spin c Dirac operator D B is defined by(4.6) D B = 1 √ n X j =1 c ( e j ) ∇ Cl e j : Ω , ∗ ( U, E ) → Ω , ∗ ( U, E ) . It is well-known that D B is formally self-adjoint (see Proposition 1.3.1 and equation (1.3.1) in [49])and D B : Ω , ± ( U, E ) → Ω , ∓ ( U, E ) .Write ∂ ∗ : Ω ,q +1 ( U, E ) → Ω ,q ( U, E ) for the adjoint of ∂ : Ω ,q ( U, E ) → Ω ,q +1 ( U, E ) with respectto the L inner product on Ω ,q ( U, E ) induced by h · , · i and h · | · i E ( q = 0 , , , . . . , n − ). Then, byTheorem 1.4.5 in [49](4.7) D B = ∂ + ∂ ∗ + A B : Ω , ± ( U, E ) → Ω , ∓ ( U, E ) where A B : Ω , ± ( U, E ) → Ω , ∓ ( U, E ) is a smooth zeroth order operator (and A B = A B ( z ) , indepen-dent of θ ). Note that A B as an operator Ω , ∗ ( U, E ) → Ω , ∗ ( U, E ) is self-adjoint because both D B and ∂ + ∂ ∗ are so.The following is instrumental in forming a global operator from local ones, whose proof is based oncanonical coordinates of BRT trivializations. Note for u ∈ Ω ,q ( D, E ) ( q = 0 , , , . . . , n ) with u = u ( z ) ,i.e. u is independent of θ , we may identify such u with an element in Ω ,q ( U, E ) by using (2.12) (andvice versa). Proposition 4.2. Let B = ( D, ( z, θ ) , ϕ ) and e B = ( D, ( w, η ) , e ϕ ) be two BRT trivializations with D = U × ] − ε, ε [ for ε > and an open set U of C n . Let A B , A e B : Ω , ± ( U, E ) → Ω , ∓ ( U, E ) be the operators EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 43 given by (4.7) . Fix an m ∈ Z . For u ∈ Ω , ± m ( X, E ) we can write u (= u | D ) = e − imθ v ( z ) = e − imη e v ( w ) forsome v ( z ) , e v ( w ) ∈ Ω , ± ( U, E ) . Then (4.8) e − imθ A B ( v ( z )) = e − imη A e B ( e v ( w )) on D. Proof. Although we shall only use part of the coordinate transformation of BRT trivializations (2.11) w = H ( z ) , ∂H ( z ) = 0; η = θ + arg g ( z ); e ϕ ( H ( z ) , H ( z )) = ϕ ( z, z ) + log | g ( z ) | , (4.9)let’s give a geometrical interpretation of how the above can be obtained for an independent inter-est. This complements the treatment of [4]. See Subsection 5.1 (cf. Remark 5.2) for its use in theconstruction of a modified Kodaira Laplacian as well as in the proof of Lemma 7.6.To see (4.9), we are going to realize D (possibly after shrinking it) as (part of) the total space of acircle bundle associated with a trivial holomorphic line bundle L over a complex manifold U ⊂ C n .More precisely suppose L is equipped with a Hermitian metric such that a local basis has || || = e − φ ( z ) and Y = { ( z, λ ) ⊂ C n +1 ; | λ | e φ = 1 } is the circle bundle inside the L ∗ . Write ρ = | λ | e φ − and λ | Y = e − φ − iθ . One has T , Y = (Ker ∂ρ ) ∩ T , C n +1 . In terms of ( z, θ ) coordinates on Y , onehas T , Y = { ∂∂z j + i ∂φ∂z j ( z ) ∂∂θ ; j = 1 , , . . . , n } (and T , Y = T , Y ) because the RHS is checked to becontained in Ker ∂ρ and it has the correct dimension. In view of Theorem 2.9 by taking the above φ to be ϕ of Z j in (2.10) and mapping ( z, θ ) of D to ( z, e − φ ( z ) − iθ ) of Y , it will be seen that D is realizedin this way as a portion of Y .For this realization, we compare the above with [4, (0.1) or Theorem I.2] and set up the (holomor-phic) transformation in coordinates: setting our ϕ , λ , θ and z (on U ) above to be φ , e iw , − s and z respectively in [4, (0.1) and Theorem I.2]). Then our Z j in (2.10) corresponds to L j in [4, PropositionI.2]. One sees that the ambient complex space of [4] is (locally) biholomorphic to part of the above L ∗ . For the next purpose, let us give an instrinsic formulation of this complex space from another pointof view. Let R > be the set of positive real numbers. Consider D × R > and equip it with the complexstructure J defined as follows. J | T U is set to be the complex structure of U ; it suffices to define J on ] − ε, ε [ × R > with coordinates s, r : J ( ∂/∂r ) = − r ∂/∂s , J ( r ∂/∂s ) = ∂/∂r . Note this definition of J is independent of choice of BRT trivializations (since Z j ⊂ T , Y identified with T , U , gives a localbasis of T , D ; see Theorem 2.9, and { z }× ] − ε, ε [ for each z ∈ U is (part of) an S orbit in X ). J isseen to be (equivalent to) the complex structure on L ∗ (with ( z, s, r ) ∈ D × R > and ( z, re is ) ∈ L ∗ incorrespondence), hence an integrable complex structure.Let now e B = ( D, ( w, η ) , e ϕ ) be any other BRT chart. Correspondingly we will denote the associatedobjects by the same notation as in B but topped with a tilde. Note that in e B the set defined by w = w for a fixed w is part of an S orbit in X ; the same can be said with B . Conversely, any S orbit of X is described (locally) by the θ parameter in any BRT charts with z -coordinates beingfixed. By using ( D × R > , J ) above, one has L ∗ ∼ = ˜ L ∗ (locally) by a biholomorphism F that preservesrespective fibers (since these are S orbits, described by θ parameters in each chart and hence must bein correspondence via D × R > by the property of BRT charts as just remarked). Further, one sees that F restricted to fibers has to be linear, hence F is a bundle isomorphism. Geometrically this pictureis essentially the same as a local change of holomorphic coordinates on the base manifold U and achange of a local basis of L ∗ by ˜ e ∗ ( z ) = g ( z ) e ∗ ( z ) with || e ∗ || = e ϕ , || ˜ e ∗ || = e ϕ for some nowherevanishing holomorphic function g ( z ) on U . The above transformation formula (4.9) easily followsfrom this concrete realization.Using the above transformation (4.9), one claims D B = D e B on Ω , ± ( U, E ) ,A B = A e B on Ω , ± ( U, E ) . (4.10)To see this for the case without E , note D is just realized as (part of) the total space of a circlebundle of a holomorphic line bundle. Clearly ∂ U + ∂ ∗ U does not depend on choice of holomorphic EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 44 coordinates on U ; that is, ∂ U + ∂ ∗ U is an intrinsic object (cf. the Hermitian metric used for ∂ ∗ U isintrinsic, (4.1)). The same idea can be applied to D B which is defined above as an intrinsic object too.Therefore (4.10) holds with the change of coordinates in (4.9) (using only w = H ( z ) ). Now with E resumed, the reasoning is basically unchanged. Hence the claim (4.10).By (4.9) v ( z ) = e − imG ( z ) e v ( w ) ( G ( z ) = arg g ( z ) ), hence by (4.10) e − imθ A B ( v ( z )) = e − imθ A B ( e − imG ( z ) e v ( w )) = e − imθ A e B ( e − imG ( z ) e v ( w )) = e − imη A e B ( e v ( w )) , proving the Proposition. (cid:3) We are now ready to introduce a global operator: Definition 4.3. For every m ∈ Z , let A m : Ω , ± m ( X, E ) → Ω , ∓ m ( X, E ) be the linear operator definedas follows. Let u ∈ Ω , ± m ( X, E ) . Then, v := A m u is an element in Ω , ∓ m ( X, E ) such that for everyBRT trivialization B := ( D, ( z, θ ) , ϕ ) ( D = U × ] − ε, ε [ , ε > , U an open set in C n ) we have v | D = e − imθ A B ( e u )( z ) where u = e − imθ e u ( z ) on D for some e u ∈ Ω , ± ( U, E ) and A B is given in (4.7).In view of Proposition 4.2, Definition 4.3 is well-defined.We are now in a position to define the modified Kohn Laplacian ( Spin c Kohn Laplacian) includinga type of CR Spin c Dirac operator e D b,m . One goal of this part is to express the index of e D b,m in anintegral form of the heat kernel density (cf. Proposition 4.6).The treatment below mostly follows traditional cases except the use of the projection operator Q ± m together with its explicit expression in integral (see (4.16) and (4.17)).By using A m in Definition 4.3 we consider e D b,m = ∂ b + ∂ ∗ b + A m : Ω , • m ( X, E ) → Ω , • m ( X, E ) , e D ± b,m = ∂ b + ∂ ∗ b + A m : Ω , ± m ( X, E ) → Ω , ∓ m ( X, E ) (4.11)with the formal adjoint e D ∗ b,m on Ω , • m ( X, E ) .We remark that e D ∗ b,m = e D b,m on Ω , • m ( X, E ) . For, by (2.5) the L inner product on Ω ,qm ( D, E ) is clearly ε ( · , · ) with the L inner product ( · , · ) on Ω ,q ( U, E ) . Now that A B is self-adjoint on Ω , ∗ ( U, E ) as aforementioned, it follows that A m is self-adjoint on Ω , • m ( X, E ) . That e D b,m is self-adjointfollows as ∂ b + ∂ ∗ b is also self-adjoint.The modified/ Spin c Kohn Laplacian is given by e (cid:3) b,m := e D ∗ b,m e D b,m : Ω , • m ( X, E ) → Ω , • m ( X, E ) e (cid:3) + b,m := e D ∗ b,m e D b,m = e D − b,m e D + b,m : Ω , + m ( X, E ) → Ω , + m ( X, E ) ( e (cid:3) − b,m := e D + b,m e D − b,m ) . (4.12)We extend e (cid:3) + b,m and e (cid:3) − b,m by e (cid:3) ± b,m : Dom e (cid:3) ± b,m ( ⊂ L , ± m ( X, E )) → L , ± m ( X, E ) (4.13)where Dom e (cid:3) ± b,m := { u ∈ L , ± m ( X, E ); distribution e (cid:3) ± b,m u ∈ L , ± m ( X, E ) } .Clearly e (cid:3) b,m − T (where T is the real vector field induced by the S action) is (the restriction of)an elliptic operator on X since ( ∂ b + ∂ ∗ b ) − T is.Put as usual, the Sobolov spaces (cf. Subsection 2.1) H s, + ( X, E ) , H s, − ( X, E ) the even and oddpart of H s ( X, T ∗ , • ⊗ E ) . In the same vein as Lemma 3.3 and Lemma 3.4 one has Dom e (cid:3) ± b,m = L , ± m ( X, E ) \ H , ± ( X, E ) , e (cid:3) + b,m and e (cid:3) − b,m are self-adjoint . (4.14)Further, for the spectrum Spec e (cid:3) + b,m ⊂ [0 , ∞ [ (resp. Spec e (cid:3) − b,m ⊂ [0 , ∞ [ ) one has (similar to Propo-sition 3.5) EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 45 Proposition 4.4. Spec e (cid:3) + b,m is a discrete subset of [0 , ∞ [ . For any µ ∈ Spec e (cid:3) + b,m , µ is an eigenvalue of e (cid:3) + b,m and the eigenspace e E + m,ν ( X, E ) := (cid:8) u ∈ Dom e (cid:3) + b,m ; e (cid:3) + b,m u = νu (cid:9) is finite dimensional with e E + m,ν ( X, E ) ⊂ Ω , + m ( X, E ) . Similar results hold for the case of (cid:3) − b,m . The following can be proved by standard argument. Lemma 4.5. We have Spec e (cid:3) + b,m T ]0 , ∞ [= Spec e (cid:3) − b,m T ]0 , ∞ [ , and for every = µ ∈ Spec e (cid:3) + b,m , dim e E + m,µ ( X, E ) = dim e E − m,µ ( X, E ) . We are going to introduce a McKean-Singer type formula (Corollary 4.8). Let F be a complex vectorbundle over X of rank r with a Hermitian metric h · | · i F . Let A ( x, y ) ∈ C ∞ ( X × X, F ⊠ F ∗ ) . For every u ∈ C ∞ ( X, F ) , R X A ( x, y ) u ( y ) dv X ( y ) ∈ C ∞ ( X, F ) is defined in a fairly standard manner.Much of what follows parallels the classical cases except that Q + m is introduced in our case. For ν ∈ Spec e (cid:3) ± b,m let P ± m,ν : L , ± ( X, E ) → e E ± m,ν ( X, E ) be the orthogonal projections (with respect to ( · | · ) E ), and P ± m,ν ( x, y ) ( ∈ C ∞ ( X × X, ( T ∗ , ± X ⊗ E ) ⊠ ( T ∗ , ± X ⊗ E ) ∗ ) ) the distribution kernels of P ± m,ν .The heat kernels of e (cid:3) + b,m and e (cid:3) − b,m are given by(4.15) e − t e (cid:3) ± b,m ( x, y ) = P ± m, ( x, y ) + X ν ∈ Spec e (cid:3) ± b,m ,ν> e − νt P ± m,ν ( x, y ) with the associated continuous operators e − t e (cid:3) ± b,m : Ω , ± ( X, E ) → Ω , ± m ( X, E ) ⊂ Ω , ± ( X, E ) . e − t e (cid:3) ± b,m isself-adjoint on Ω , ± ( X, E ) .Remark that the heat kernels (4.15) are smooth. For, the eigenfunctions involved (in the equivalentform as (1.67)) are still eigenfunctions of ( e (cid:3) b,m − T ) hence eigenfunctions of an elliptic operator. Inthe elliptic case, one has the G¨arding type inequality which estimates the various Sobolev norms ofthe eigenfunctions, and hence mainly by Sobolev embeddings, gives eventually the smoothness of theheat kernels (cf. [37, Lemmas 1.6.3 and 1.6.5]).An important operator is given by the orthogonal projection(4.16) Q ± m : L , ± ( X, E ) → L , ± m ( X, E ) (for the m -th Fourier component). Fourier analysis with (2.2) gives(4.17) Q ± m u = 12 π Z π − π u ( e − iθ ◦ x ) e imθ dθ, ∀ u ∈ Ω , ± ( X, E ) (where u ( e − iθ ◦ x ) stands for the pull back ( e − iθ ) ∗ u , cf. (2.2)). The explicit expression (4.17) turnsout to be crucial to many (unconventional) estimates later.It is fairly standard (note Q + m in the second line below) to obtain (by (4.15)) ( ∂∂t + e (cid:3) ± b,m )( e − t e (cid:3) ± b,m u ) = 0 , ∀ u ∈ Ω , ± ( X, E ) , ∀ t > , lim t → + ( e − t e (cid:3) ± b,m u ) = Q ± m u, ∀ u ∈ Ω , ± ( X, E ) . (4.18)For ν ∈ Spec e (cid:3) + b,m , let (cid:8) f ν , . . . , f νd ν (cid:9) be an orthonormal basis for e E + m,ν ( X, E ) . Define(4.19) Tr P + m,ν ( x, x ) := d ν X j =1 (cid:12)(cid:12) f νj ( x ) (cid:12)(cid:12) E ∈ C ∞ ( X ) which is equal to Tr P + m,ν ( x, x ) = P dj =1 h P + m,ν ( x, x ) e j ( x ) | e j ( x ) i E where { e j ( x ) } j is any orthonormalbasis of T ∗ , + x X ⊗ E x . Define Tr P − m,µ ( x, x ) similarly. EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 46 Clearly d ± ν = R X Tr P ± m,ν ( x, x ) dv X ( x ) .Put Tr e − t e (cid:3) ± b,m ( x, x ) := Tr P ± m, ( x, x ) + P ν ∈ Spec e (cid:3) ± b,m ,ν> e − νt Tr P ± m,ν ( x, x ) , so for t > (4.20) Z X Tr e − t e (cid:3) ± b,m ( x, x ) dv X ( x ) = dim e E ± m, ( X, E ) + X ν ∈ Spec e (cid:3) ± b,m ,ν> e − νt dim e E ± m,ν ( X, E ) . Combining Lemma 4.5 and (4.20) gives(4.21) Z X (cid:16) Tr e − t e (cid:3) + b,m ( x, x ) − Tr e − t e (cid:3) − b,m ( x, x ) (cid:17) dv X ( x ) = dim Ker e (cid:3) + b,m − dim Ker e (cid:3) − b,m . As in Theorem 3.8 one has(4.22) Ker e D b,m = Ker e (cid:3) + b,m ⊂ Ω , + m ( X, E ) , Ker e D ∗ b,m = Ker e (cid:3) − b,m ⊂ Ω , − m ( X, E ) . Put ind e D b,m := dim Ker e D b,m − dim Ker e D ∗ b,m . We express the index (by (4.22) and (4.21)) as Proposition 4.6. For every t > , we have (4.23) ind e D b,m = Z X (cid:16) Tr e − t e (cid:3) + b,m ( x, x ) − Tr e − t e (cid:3) − b,m ( x, x ) (cid:17) dv X ( x ) . The invariance of the index is expressed by the following (some aspects on ind D b,m refer to Theo-rem 3.8). Theorem 4.7. (Homotopy invariance) We have ind D b,m = ind e D b,m . To summarize (with Theorem 4.7, Proposition 4.6 and (3.13)) we have a McKean-Singer formula(cf. Corollary 5.15 for McKean-Singer (II)). Corollary 4.8. (McKean-Singer (I)) Fix m ∈ Z . For t > , we have (4.24) n X j =0 ( − j dim H jb,m ( X, E ) = Z X (cid:16) Tr e − t e (cid:3) + b,m ( x, x ) − Tr e − t e (cid:3) − b,m ( x, x ) (cid:17) dv X ( x ) . Remark . To compare with the original Kohn Laplacian, a similar formula (as Corollary 4.8) n X j =0 ( − j dim H jb,m ( X, E ) = Z X (cid:16) Tr e − t (cid:3) + b,m ( x, x ) − Tr e − t (cid:3) − b,m ( x, x ) (cid:17) dv X ( x ) holds. When X is not CR K¨ahler, it is obscure, by the experience from classical cases, to calculate thedensity Tr e − t (cid:3) + b,m ( x, x ) − Tr e − t (cid:3) − b,m ( x, x ) with the original Kohn Laplacian. The introduction of themodified Kohn Laplacians replacing (cid:3) ± b,m by e (cid:3) ± b,m is expected to facilitate this calculation. But becauseof the unconventional asymptotic expansion of e − t e (cid:3) ± b,m ( x, x ) some novelty beyond the classical casesshows up (as mentioned in Introducton). It should be noted that when X is CR K¨ahler, e (cid:3) ± b,m = (cid:3) ± b,m .To prove Theorem 4.7 in the remaining of this section, observe that it is nothing but a statementof homotopy invariance of index. For, with A m a global operator (see Definition 4.3), putting L t = ∂ b,m + ∂ ∗ b,m + tA m : Ω , + m ( X, E ) → Ω , − m ( X, E ) for t ∈ [0 , , gives the homotopy between L = D b,m and L = e D b,m .Remark that there have been proofs for results of this type; for instance, see [6] using heat kernelmethod and [2] using functional analysis method (both not exactly formulated in the above formthough). To make it accessible to a wider readership, we include a (comparatively) self-contained andshort proof. It is amusing to note that the Hodge theory in Section 3 is useful at certain points of ourproof.Some preparations are in order. We extend L t by setting Dom L t = { u ∈ L , + m ( X, E ); distribution L t u ∈ L , − m ( X, E ) } EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 47 so that L t : Dom L t ( ⊂ L , + m ( X, E )) → L , − m ( X, E ) . Write L ∗ t for the Hilbert space adjoint of L t .Let H , + m ( X, E ) be the completion of Ω , + m ( X, E ) with respect to the Hermitian inner product Q ( u, v ) = ( u | v ) E + ( ∂ b u | ∂ b v ) E + ( ∂ ∗ b u | ∂ ∗ b v ) E . Clearly H , + m ( X, E ) ⊂ Dom L t , ∀ t ∈ R . One can show that H , + m ( X, E ) = Dom L t . Let t = 0 . Assume L f = u with f, u ∈ L , + m and also assume f ⊥ Ker (cid:3) + b,m since for any smooth g , f + g ∈ H , + m iff f ∈ H , + m . Using the partial inverse N m in (3.2) of our Hodge theory in Section 3, we have L N m f = N m L f = N m u since L commutes with N m as in the ordinary Hodge theory. Now L ∗ L N m f = (cid:3) + b,m N m f = f by (3.2), one has f = L ∗ N m u . But N m u increases the (Sobolev) order of regularity of u by and then L ∗ N m u decreases by 1, the regularity of f is of order 1. By localization, with a partitionof unity, on an open subset D in place of X and by the formula (2.14) in BRT charts D = U × ] − δ, δ [ for ∂ b , it follows from the standard G¨arding’s inequality (e.g. [38, p. 93]) that the above Q ( · , · ) isequivalent to the Sobolev norm of order one (on the m -th component). Hence f ∈ H , + m . For t = 0 ,since L t = L + tA m with A m a smooth zeroth order operator, it follows Dom L t = Dom L .Consider H := H , + m ( X, E ) ⊕ Ker L ∗ and H = L , − m ( X, E ) ⊕ Ker L . Let ( · | · ) H and ( · | · ) H beinner products on H and H respectively, given by ( ( f , g ) | ( f , g ) ) H = Q ( f , f ) + ( g | g ) E , ( ( e f , e g ) | ( e f , e g ) ) H = ( e f | e f ) E + ( e g | e g ) E . Let P Ker L denote the orthogonal projection onto Ker L with respect to ( · | · ) E .Let A t : H → H be the (continous) linear map defined as follows. For ( u, v ) ∈ H , A t ( u, v ) = ( L t u + v, P Ker L u ) ∈ H . Lemma 4.10. There is a r > such that A t : H → H is invertible, for every ≤ t ≤ r .Proof. We first claim that(4.25) A is invertible . If A ( u, v ) = 0 for some ( u, v ) ∈ H , then(4.26) i ) L u = − v ∈ Ker L ∗ , ii ) P Ker L u = 0 . By (4.26)(4.27) ( L u | L u ) E = − ( L u | v ) E = − ( u | L ∗ v ) E = 0 , giving u ∈ Ker L . Hence by ii) of (4.26) we obtain u = 0 , giving also v = 0 by i) of (4.26). We haveproved that A is injective.We shall now prove that A is surjective. Let ( a, b ) ∈ H . First we note L : Dom L → L , − m ( X, E ) has an L closed range, so(4.28) a = L α + β, α ∈ H , + m ( X, E ) , α ⊥ Ker L , β ∈ (Rang L ) ⊥ = Ker L ∗ . Another way to see (4.28) is to use (cid:3) − b,m N − m + Π − m = I (on L − m ) of (3.2) (for the “ − ” case) of Hodgetheory in Section 3, and obtain a = L L ∗ N − m a + Π − m a where Π − m a ∈ Ker L ∗ (cf. Theorem 3.8) and L ∗ N − m a ∈ H , + m ( X, E ) as mentioned above this lemma. In either way, by (4.28) one sees A ( α + b, β ) = ( L ( α + b ) + β, P Ker L ( α + b )) = ( L α + β, b ) = ( a, b ) . Thus A is surjective. The claim (4.25) follows.Let A − : H → H be the inverse of A . It follows from open mapping theorem that A − iscontinuous. EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 48 To finish the proof the following arguement based on geometric series is standard. Write A t = A + R t , where R t : H → H is continuous and there is a constant c > such that k R t u k H ≤ ct k u k H ,for u ∈ H . Put H t = I − A − R t + ( A − R t ) − ( A − R t ) + · · · , e H t = I − R t A − + ( R t A − ) − ( R t A − ) + · · · . Since A − is continuous, H t : H → H and e H t : H → H are well-defined as continuous maps forsmall t ≥ . Moreover A t ◦ ( H t ◦ A − ) = I (on H ) and ( A − ◦ e H t ) ◦ A t = I on ( H ), giving right andleft inverses of A t for small t ≥ . Hence the lemma. (cid:3) For t ∈ [0 , write L ∗ t : Dom L ∗ t ( ⊂ L , − m ( X, E )) → L , + m ( X, E ) for the adjoint of L t with respectto ( · | · ) E . Similar to L and L , one has dim Ker L t < ∞ and dim Ker L ∗ t < ∞ (with Ker L t ⊂ Ω , + m ( X, E ) , Ker L ∗ t ⊂ Ω , − m ( X, E ) ).Put ind L t := dim Ker L t − dim Ker L ∗ t . Lemma 4.11. There is a r > such that ind L t = ind L , for every ≤ t ≤ r .Proof. Let r > be as in Lemma 4.10. We first show that(4.29) ind L ≤ ind L t , ∀ ≤ t ≤ r. Fix ≤ t ≤ r . We define B : Ker L ∗ t ⊕ Ker L → Ker L t ⊕ Ker L ∗ as follows. Let ( a, b ) ∈ Ker L ∗ t ⊕ Ker L . By Lemma 4.10, A t : H , + m ( X, E ) ⊕ Ker L ∗ → L , − m ( X, E ) ⊕ Ker L is invertible. There is a unique ( u, v ) ∈ H , + m ( X, E ) ⊕ Ker L ∗ = Dom L t ⊕ Ker L ∗ such that A t ( u, v ) =( a, b ) . Let P Ker L t : L , + m ( X, E ) → Ker L t be the orthogonal projection with respect to ( · | · ) E . Thenthe above map B is defined by B ( a, b ) := ( P Ker L t u, v ) ∈ Ker L t ⊕ Ker L ∗ . We claim that B is injective. If so, then(4.30) dim Ker L ∗ t + dim Ker L ≤ dim Ker L t + dim Ker L ∗ , i.e. dim Ker L − dim Ker L ∗ ≤ dim Ker L t − dim Ker L ∗ t , yielding the desired (4.29).For the claim that B is injective, if B ( a, b ) = (0 , for some ( a, b ) ∈ Ker L ∗ t ⊕ Ker L , write ( u, v ) ( ∈ H , + m ( X, E ) ⊕ Ker L ∗ ) such that A t ( u, v ) = ( a, b ) . As (0 , 0) = B ( a, b ) = ( P Ker L t u, v ) , P Ker L t u = 0 and v = 0 . Using the definition of A t , one has(4.31) A t ( u, v ) = A t ( u, 0) = ( L t u, P Ker L u ) = ( a, b ) ∈ Ker L ∗ t ⊕ Ker L to give a = L t u ∈ Ker L ∗ t , hence ( a | a ) E = ( a | L t u ) E = ( L ∗ t a | u ) E = 0 gives L t u = a = 0 so that u ∈ Ker L t , i.e. u = P Ker L t u by definition. It follows that u = 0 since P Ker L t u = 0 as just seen. With u = 0 and (4.31) one sees ( a, b ) = ( L t u, P Ker L u ) = (0 , , giving the injectivity of B .By the same argument, ind L ∗ ≤ ind L ∗ t for small t . By ind L ∗ t = − ind L t , ind L ≥ ind L t holds. Thisand (4.29) prove the lemma. (cid:3) Proof of Theorem 4.7. Let I := { r ∈ [0 , there is an ε > such that ind L t = ind L , ∀ t ∈ ( r − ε, r + ε ) T [0 , } .I = ∅ is open by Lemma 4.11. Around a limit point r ∞ of I , by the same type of argument in theproof of Lemma 4.11 and Lemma 4.10 (replacing t = 0 by t = r ∞ in H , H and A ), one finds ind L t = ind L r ∞ for t ∈ ( r ∞ − ε , r ∞ + ε ) with some ǫ > . This implies I is closed in [0 , , so I = [0 , . (cid:3) EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 49 5. A SYMPTOTIC EXPANSIONS FOR THE HEAT KERNELS OF THE MODIFIED K OHN L APLACIANS In view of the McKean-Singer formula (Corollary 4.8), one of the goals is to calculate the localdensity (i.e. the term to the right of (4.24)). It consists in obtaining an asymptotic expansion for theheat kernel of the modified Kohn Laplacian ( Spin c Kohn Laplacian), to which we base our approachon two main steps. While the first step is motivated by the globally free case (see Theorem 1.2), itwill be replaced by a local version within the framework of BRT trivializations (Section 2.4). A crucialoff-diagonal estimate is going to be done in this subsection (cf. Theorem 5.9). In the second stepwe use the adjoint version of the heat equation to construct a global heat kernel with an asymptoticexpansion related to local kernels .5.1. Heat kernels of the modified Kodaira Laplacians on BRT trivializations. This subsection ismotivated by the globally free case (cf. Theorem 1.2). Here the emphasis is made on the localizationof the argument including the Spin c structure (which is needed for explicit local formulas of the heatkernel density). An important heat kernel estimate, termed as off-diagonal estimate , will be establishedin Theorem 5.9.It is worth remarking that in the statement and proof of Theorem 1.2, we make no use of harmonictheory. In the locally free case, by contrast, it will be an important step to relate the (modified) KohnLaplacian to (modified) Kodaira Laplacian (see discussion after that theorem). Since these Laplaciansare defined via certain adjoints , suitable matching of metrics involved in both Laplacians must be doneas an essential step.We will use the same notations as in Section 4. Let B := ( D, ( z, θ ) , ϕ ) be a BRT trivialization (with D = U × ] − ε, ε [ , ε > and U an open subset of C n , cf. Subsection 2.4). For x ∈ D wrtie z = z ( x ) and θ = θ ( x ) . Since E is rigid and CR, equipped with a rigid Hermitian (fiber) metric h · | · i E , (as inSection 4) E descends as a (holomorphically trivial) vector bundle over U (possibly after shrinking U )and h · | · i E as a Hermitian (fiber) metric on E over U .Let L → U be a trivial (complex) line bundle with a non-trivial Hermitian fiber metric | | h L = e − ϕ ( ϕ as in the above BRT triple B ). Write ( L m , h L m ) → U for the m -th power of ( L, h L ) . Let Ω ,q ( U, E ⊗ L m ) be the space of (0 , q ) forms on U with values in E ⊗ L m ( q = 0 , , , . . . , n ). As usual, Ω , + ( U, E ⊗ L m ) and Ω , − ( U, E ⊗ L m ) denote forms of even and odd degree.To start with the matching of the metrics we let h · , · i be the Hermitian metric on C T U given by(cf. (4.1))(5.1) h ∂∂z j , ∂∂z k i = h ∂∂z j − i ∂ϕ∂z j ( z ) ∂∂θ | ∂∂z k − i ∂ϕ∂z k ( z ) ∂∂θ i , j, k = 1 , , . . . , n. h · , · i induces Hermitian metrics on T ∗ ,q U (bundle of (0 , q ) forms on U ), denoted also by h · , · i .These metrics induce Hermitian metrics on T ∗ ,q U ⊗ E , still denoted by h · | · i E .Let ( · , · ) be the L inner product on Ω ,q ( U, E ) induced by h · , · i , h · | · i E , and similarly ( · , · ) m the L inner product on Ω ,q ( U, E ⊗ L m ) induced by h · , · i , h · | · i E and h L m .Let ∂ L m : Ω ,q ( U, E ⊗ L m ) → Ω ,q +1 ( U, E ⊗ L m ) , ( q = 0 , , , . . . , n − , be the Cauchy-Riemannoperator. Let ∂ ∗ L m : Ω ,q +1 ( U, E ⊗ L m ) → Ω ,q ( U, E ⊗ L m ) be the formal adjoint of ∂ L m with respect to ( · , · ) m .An essential operator that enters our picture is the following one (of Dirac type).(5.2) D B,m (= D + − B,m ) := ∂ L m + ∂ ∗ L m + A B : Ω , + ( U, E ⊗ L m ) → Ω , − ( U, E ⊗ L m ) where A B : Ω , + ( U, E ⊗ L m ) → Ω , − ( U, E ⊗ L m ) is as in (4.7) (replacing E there by E ⊗ L m here)and(5.3) D ∗ B,m : Ω , − ( U, E ⊗ L m ) → Ω , + ( U, E ⊗ L m ) EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 50 the formal adjoint of D B,m with respect to ( · , · ) m . (Note D B,m on the full Ω , ∗ = Ω , + ⊕ Ω , − isself-adjoint; see the line below (4.6). But we prefer to use the above D ∗ B,m in the present context.)Note also L with the metric h L depends on the choice of a BRT trivialization. However, A B is indeedan intrinsic object; we refer to Remark 5.2 in this regard.One has the modified/ Spin c Kodaira Laplacian :(5.4) e (cid:3) + B,m := D ∗ B,m D B,m : Ω , + ( U, E ⊗ L m ) → Ω , + ( U, E ⊗ L m ) . One may define e (cid:3) − B,m : Ω , − ( U, E ⊗ L m ) → Ω , − ( U, E ⊗ L m ) analogously (by starting with D − + B,m or D ∗ B,m ).The following fact appears fundamental in itself. It is instrumental to our construction of a heatkernel (cf. (5.44)) (See, however, remarks after its proof). Proposition 5.1. In notations above let u ∈ Ω , ± m ( X, E ) . On D we can write u ( z, θ ) = e − imθ e u ( z ) forsome e u ( z ) ∈ Ω , ± ( U, E ) . Recall the modified Kohn Laplacian e (cid:3) ± b,m in (4.12) . We write s for the localbasis m of L m . Then (5.5) e − mϕ e (cid:3) ± B,m ( e mϕ e u ⊗ s ) = ( e imθ e (cid:3) ± b,m ( u )) ⊗ s. Without the danger of confusion we may write (5.6) e − mϕ e (cid:3) ± B,m ( e mϕ e u ) = e imθ e (cid:3) ± b,m ( u ) . Proof. One may work out this result by explicit computations. The following gives a somewhat con-ceptual proof. The idea is that one continues to match the objects on U and on D ( ⊂ X ) . (In this wayit turns out that no explicit computations of these Laplacians in local coordinates are needed.)We define χ (= χ q ) : Ω ,q ( U, E ) → Ω ,q ( U, E ⊗ L m ) ( q = 0 , , , . . . , n ) by ˜ v ( z ) → ˜ v ( z ) e mϕ ( z ) ⊗ s ( z ) for ˜ v ∈ Ω ,q ( U, E ) . Note χ preserves the (pointwise) norms. Equivalently χ ( e − mϕ ˜ v ) = ˜ v ⊗ s .We define δ ˜ v = ∂ ˜ v + m ( ∂ϕ ) ∧ ˜ v for ˜ v ∈ Ω ,q ( U, E ) where ∂ : Ω ,q ( U, E ) → Ω ,q +1 ( U, E ) . One mayverify(5.7) ∂ L m ◦ χ = χ ◦ δ on Ω ,q ( U, E ) . Indeed, by χ ( e − mϕ ∂ ˜ u ) = ∂ ˜ u ⊗ s = ∂ L m (˜ u ⊗ s ) , one sees the term to the left of (5.7): ∂ L m ◦ χ ( e − mϕ ˜ u ) (= ∂ L m (˜ u ⊗ s )) = χ ( e − mϕ ∂ ˜ u ) . Further, by using definition of δ one computes e mϕ δ ( e − mϕ ˜ u ) = ∂ ˜ u . Then χ ( e − mϕ ∂ ˜ u ) = χ ( e − mϕ ( e mϕ δ ( e − mϕ ˜ u ))) which is χ ( δ ( e − mϕ ˜ u )) , giving the term to the right of(5.7) and proving (5.7).Since χ is norm-preserving, we have also(5.8) ∂ ∗ L m ◦ χ = χ ◦ δ ∗ between respective adjoints. Combining (5.7) and (5.8) gives for (cid:3) B,m ≡ ( ∂ ∗ L m + ∂ L m ) and ∆ ≡ ( δ ∗ + δ ) (5.9) (cid:3) B,m ◦ χ = χ ◦ ∆ . By (2.14) for ∂ b , one computes, for g = e − imθ ˜ g ∈ Ω ,qm ( U × ] − ε, ε [ , E ) = Ω ,qm ( D, E ) (5.10) e imθ ∂ b ( e − imθ ˜ g ( z )) = δ (˜ g ( z )) . Write the map χ : Ω ,qm ( D, E ) → Ω ,q ( U, E ) for χ ( g ) = χ ( e − imθ ˜ g ) = ˜ g , equivalently, χ ( g ) = e imθ g .Note χ preserves the respective (pointwise) norms (cf. (5.1)).By (5.10) one sees (with ∂ b,m = ∂ b | Ω ,qm )(5.11) χ ◦ ∂ b,m = δ ◦ χ on Ω ,qm ( D, E ) . EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 51 By (2.5) the L inner product on Ω ,qm ( D, E ) is clearly ε ( · , · ) with the L inner product ( · , · ) on Ω ,q ( U, E ) . Thus, in the same way as (5.9) by using (5.11) we have for (cid:3) b,m ≡ ( ∂ ∗ b,m + ∂ b,m ) (and ∆ ≡ ( δ ∗ + δ ) as above)(5.12) χ ◦ (cid:3) b,m = ∆ ◦ χ on Ω ,qm ( D, E ) . Combining (5.12) and (5.9) yields (cid:3) b,m = ( χχ ) − ◦ (cid:3) B,m ◦ ( χχ ) (5.13)By χχ ( e − imθ ˜ u ) = e mϕ ˜ u ⊗ s and ( χχ ) − (˜ v ⊗ s ) = e − imθ e − mϕ ˜ v , one obtains(5.14) ( (cid:3) b,m u ) ⊗ s = e − imθ e − mϕ ( (cid:3) B,m e mϕ (˜ u ⊗ s )) for u = e − imθ ˜ u ∈ Ω ,qm ( D, E ) , giving e − mϕ (cid:3) B,m ( e mϕ e u ) = e imθ (cid:3) b,m ( u ) in notation similar to (5.6).For modified Laplacians, from the definition of the zeroth order operator A m : Ω , + m ( X, E ) → Ω , − m ( X, E ) (see Definition 4.3), it is clear that (in notation similar to (5.6))(5.15) e − mϕ A B ( e mϕ e u ) = e imθ A m ( u ) . In a way similar to (5.14) it follows by using (5.15) that e − mϕ D B,m ( e mϕ e u ) = e imθ e D b,m ( u ) hence easily that e − mϕ e (cid:3) + B,m ( e mϕ e u ) = e imθ e (cid:3) + b,m ( u ) proving the proposition. (cid:3) Remark that one might be led by Proposition 5.1 to reduce the study of Kohn e (cid:3) ± b,m to that of Kodaira e (cid:3) B,m . Indeed such a reduction works quite well in the globally free case (see discussion followingTheorem 1.2 in Introduction). In the locally free case (of S action), however, a naive thought ofusing the Kodaira Laplacian and its associated (local) heat kernels for a better understanding of theheat kernel in Kohn’s case is not directly accessible (see remarks following proof of Theorem 5.14).Namely the associated heat kernels of the two Laplacians cannot be easily linked as (5.5) seems tosuggest. This reflects the fact that the associated heat kernels, rather than Laplacians themselves,are objects which are more global in nature. More in this regard will be pursued in the comingSubsection 5.2 and Section 6. Remark . The definition of A B in (5.2) depends on a BRT triple, and the same can be said withProposition 5.1. To see that A B has an intrinsic meaning, one uses the transformation of BRT coordi-nates as shown in the proof of Proposition 4.2. The geometrical construction given there shows thatlocally X is part of a circle bundle inside the L ∗ (with metric induced by that of L ) over U , and thequantities such as ϕ , z and θ in a BRT triple are associated with geometric ones as metric for a localbasis (of L ), coordinates on the base U and (part of) a holomorphic coordinate on fibers (of L ∗ ) re-spectively. The transformation in these quantities with another choice of a BRT chart is nothing morethan a change of holomorphic coordinates of the same line bundle. It follows that A B is intrinsic in aproper sense. A similar explanation can be given to Proposition 5.1 too (although we do not strictlyneed this intrinsic property in what follows). Remark . In the case of certain Riemannian foliations, it is known that the Laplacian downstairsand Laplacian upstairs (in a suitable generalized sense) can be related in spirit similar to that in ourproposition above. See [57, p. 2310-2311].As remarked in Subsection 1.7.3, to suit our purpose we will actually be considering adjoint heatequation and adjoint heat kernel first.To proceed further, some notations are in order. Let M be a C ∞ orientable paracompact manifoldwith a vector bundle F over it. EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 52 Definition 5.4. Let A ( t, x ) ∈ C ∞ ( R + × M, F ) . We write A ( t, x ) ∼ t k b − k ( x, t ) + t k +1 b − k +1 ( x, t ) + t k +2 b − k +2 ( x, t ) + · · · as t → + ,b s ( x, t ) ∈ C ∞ ( M, F ) a possibly t -dependent smooth function , s = − k, − k + 1 , − k + 2 , . . . , for k ∈ Z , provided that for every compact set K ⋐ M , every ℓ, M ∈ N with M ≥ M ( m ) for some M ( m ) ( m = dim M ), there are C ℓ,K,M > , ε > and M ( m, ℓ ) (independent of t ) such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A ( t, x ) − M X j =0 t k + j b − k + j ( x, t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C ℓ ( K ) ≤ C ℓ,K,M t M − M ( m,ℓ ) , ∀ < t < ε . Remark . In the important case of the heat kernel p t ( x, y ) of a generalized Laplacian on a compactRiemannian manifold B of dimension β , M = B × B is of dimension m = 2 β and k = − β . One cantake M ( m ) = [ β ] + 1 and M ( m, ℓ ) = β + ℓ . See [6, Theorem 2.30]. In this case b s ( t, x ) for all s canbe taken to be independent of t .The novelty above is that b s could be nontrivially dependent on t (in contrast to the conventionalcase of an asymptotic expansion for heat kernels).Let T ∗ , + U and T ∗ , − U denote forms of even degree and odd degree in T ∗ , • U , respectively asbefore. If T ( z, w ) ∈ ( T ∗ , + U ⊗ E ) ⊠ ( T ∗ , + U ⊗ E ) ∗ | ( z,w ) , write | T ( z, w ) | for the standard pointwisematrix norm of T ( z, w ) induced by h · , · i and h · | · i E .Suppose G ( t, z, w ) ∈ C ∞ ( R + × U × U, ( T ∗ , + U ⊗ E ) ⊠ ( T ∗ , + U ⊗ E ) ∗ ) . As usual, we denote G ( t ) : Ω , +0 ( U, E ) → Ω , + ( U, E ) (resp. G ′ ( t ) ) the continuous operator associated with the kernel G ( t, z, w ) (resp. ∂G ( t,z,w ) ∂t ) ( Ω , +0 ( U, E ) denotes elements of compact support in U ).We are now ready to consider the heat operators associated with e (cid:3) + B,m and e (cid:3) − B,m in an adjoint version. By using the Dirichlet heat kernel construction (see [36] or [14]) we can obtain the theoremstated in the following form. Proposition 5.6. There exists an A B, + ,m ( t, z, w ) =: A B, + ( t, z, w ) ∈ C ∞ ( R + × U × U, ( T ∗ , + U ⊗ E ) ⊠ ( T ∗ , + U ⊗ E ) ∗ ) such that lim t → A B, + ( t ) = I in D ′ ( U, T ∗ , + U ⊗ E ) ,A ′ B, + ( t ) u + A B, + ( t )( e (cid:3) + B,m u ) = 0 , ∀ u ∈ Ω , +0 ( U, E ) , ∀ t > , (5.16) and A B, + ( t, z, w ) satisfies the following: (I) For every compact set K ⋐ U and every α , α , β , β ∈ N n ,every γ ∈ N , there are constants C γ,α ,α ,β ,β ,K > , ε > and P ∈ N independent of t such that (5.17) (cid:12)(cid:12)(cid:12) ∂ γt ∂ α z ∂ α z ∂ β w ∂ β w A B, + ( t, z, w ) (cid:12)(cid:12)(cid:12) ≤ C γ,α ,α ,β ,β ,K t − P e − ε | z − w | t , ∀ ( t, z, w ) ∈ R + × K × K. (II) Let g ∈ Ω , +0 ( U, E ) . For every α , α ∈ N n and every compact set K ⋐ U , there is a C α ,α ,K > independent of t such that sup {| ∂ α z ∂ α z ( A B, + ( t ) g )( z ) | ; z ∈ K }≤ C α ,α ,K X β ,β ∈ N n , | β | + | β |≤| α | + | α | sup n(cid:12)(cid:12)(cid:12) ∂ β z ∂ β z g ( z ) (cid:12)(cid:12)(cid:12) ; z ∈ U o . (5.18) (III) A B, + ( t, z, w ) admits an asymptotic expansion in the following sense (see Definition 5.4 for ∼ ). Forsome K B, + ( t, z, w ) A B, + ( t, z, w ) = e − h +( z,w ) t K B, + ( t, z, w ) ,K B, + ( t, z, w ) ∼ t − n b + n ( z, w ) + t − n +1 b + n − ( z, w ) + · · · + b +0 ( z, w ) + tb + − ( z, w ) + · · · as t → + ,b + s ( z, w ) (= b + s,m ( z, w )) ∈ C ∞ ( U × U, ( T ∗ , + U ⊗ E ) ⊠ ( T ∗ , + U ⊗ E ) ∗ ) , s = n, n − , n − , . . . , (5.19) EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 53 where h + ( z, w ) ∈ C ∞ ( U × U, R + ) with h + ( z, z ) = 0 for every z ∈ U and for every compact set K ⋐ U ,there is a constant C K > such that C k | z − w | ≤ h + ( z, w ) ≤ C K | z − w | .In (5.16) with e (cid:3) − B,m in place of e (cid:3) + B,m , corresponding statements (with A B, − , K B, − etc.) for e (cid:3) − B,m holdtrue as well.Remark . One may use a (weaker) version in the sense of asymptotic heat kernel (cf. [23], p. 96for an informal explanation) with the property that this kernel (without uniqueness) satisfies (5.17)-(5.19) (so that it admits the same asymptotic expansion as the above kernel) and also satisfies (5.16) asymptotically (cf. Lemma 5.11 with (5.43)). Although the asymptotic heat kernels are not unique,the “formal” heat kernel given (in above notation) by this form e − h +( z,w ) t ( t − n b + n ( z, w ) + t − n +1 b + n − ( z, w ) + · · · + b +0 ( z, w ) + tb + − ( z, w ) + · · · ) is unique.We are interested in calculating Tr b + s ( z, z ) − Tr b − s ( z, z ) ( s = n, n − , . . . , ) (where Tr b ± s ( z, z ) = P j h b + s ( z, z ) e j | e j i E for any orthonormal frame e j of T ∗ , + z U ⊗ E z ). The idea relies on Lichnerowiczformulas for (modified/ Spin c Kodaira Laplacians) e (cid:3) + B,m and e (cid:3) − B,m (cf. Theorem 1.3.5 and Theorem1.4.5 in [49]) so that the (by now standard) rescaling technique in [6] and [23] can apply well.To state the result precisely, we introduce some notations. Let ∇ T U be the Levi-Civita connectionon C T U with respect to h · , · i . Let P T , U be the natural projection from C T U onto T , U . ∇ T , U := P T , U ∇ T U is a connection on T , U . Let ∇ E ⊗ L m be the (Chern) connection on E ⊗ L m → U (inducedby h · , · i E and h L m , see Theorem 2.12). Let Θ( ∇ T , U , T , U ) ( ∈ C ∞ ( U, Λ ( C T ∗ U ) ⊗ End ( T , U ))) and Θ( ∇ E ⊗ L m , E ⊗ L m ) ( ∈ C ∞ ( U, Λ ( C T ∗ U ) ⊗ End ( E ⊗ L m ))) be the associated curvatures. As incomplex geometry, put Td ( ∇ T , U , T , U ) = e Tr ( h ( i π Θ( ∇ T , U ,T , U ))) , h ( z ) = log( z − e − z ) , ch ( ∇ E ⊗ L m , E ⊗ L m ) = Tr ( e h ( i π Θ( ∇ E ⊗ L m , E ⊗ L m ))) , e h ( z ) = e z . Then the above calculation leads to the following. (cid:16) Tr b + s ( z, z ) − Tr b − s ( z, z ) (cid:17) = 0 , s = n, n − , . . . , , (cid:16) Tr b +0 ( z, z ) − Tr b − ( z, z ) (cid:17) dv U ( z ) = h Td ( ∇ T , U , T , U ) ∧ ch ( ∇ E ⊗ L m , E ⊗ L m ) i n ( z ) , ∀ z ∈ U, (5.20)where [ · · · ] | n denotes the n -form part.As the calculation to be performed here is almost entirely the same as in the standard case, we omitthe detail.Let ∇ T X be the Levi-Civita connection on T X with respect to h · | · i and ∇ E the connection on E associated with h · | · i E (cf. Theorem 2.12). In similar notation as above ∇ T , X := P T , X ∇ T X is aconnection on T , X .Since ∇ T , X and ∇ E are rigid, in view of compatibility of metrics (and connections) in (5.1) andTheorem 2.12, one sees that ∀ ( z, θ ) ∈ D (for ω see lines below Definition 2.1): Td ( ∇ T , U , T , U )( z ) = Td b ( ∇ T , X , T , X )( z, θ )ch ( ∇ E ⊗ L m , E ⊗ L m )( z ) = (cid:16) ch b ( ∇ E , E ) ∧ e − m dω π (cid:17) ( z, θ ) (5.21)and h Td ( ∇ T , U , T , U ) ∧ ch ( ∇ E ⊗ L m , E ⊗ L m ) i n ( z ) ∧ dθ = h Td b ( ∇ T , X , T , X ) ∧ ch b ( ∇ E , E ) ∧ e − m dω π ∧ ω i n +1 ( z, θ ) . (5.22) EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 54 To sum up we arrive at the following (by (5.20), (5.22) and dv U ∧ dθ = dv X on D cf. (2.5)) Proposition 5.8. With the notations above, we have (cid:16) Tr b +0 ( z, z ) − Tr b − ( z, z ) (cid:17) dv X ( z, θ )= h Td b ( ∇ T , X , T , X ) ∧ ch b ( ∇ E , E ) ∧ e − m dω π ∧ ω i n +1 ( z, θ ) , ∀ ( z, θ ) ∈ D. (5.23)To state the final technical result of this subsection, we first identify T ∗ , • z U ⊗ E z with T ∗ , • z U ⊗ E z (by parallel transport along geodesics joining z , z ∈ U ), so we can identify T ∈ ( T ∗ , • U ⊗ E ) ⊠ ( T ∗ , • U ⊗ E ) ∗ | ( z ,z ) with an element in End ( T ∗ , • z U ⊗ E z ) . With this identification, write(5.24) Tr z T := d X j =1 h T e j | e j i E , where e , . . . , e d is an orthonormal frame of T ∗ , • z U ⊗ E z .In the proof of Theorem 1.10 (see Theorem 6.4), somewhat surprisingly, as deviated from theclassical case, we need to estimate the off-diagonal terms Tr z b + s ( z, w ) − Tr z b − s ( z, w ) for each s . Forthis, the following can be considered as another application of the rescaling technique (and an identityin Berenzin integral as usual). Theorem 5.9. (Off-diagonal estimate) With the notations above, we have Tr z b + s ( z, w ) − Tr z b − s ( z, w ) = O ( | z − w | s ) locally uniformly on U × U , s = n, n − , . . . , . (5.25) Proof. Recall E is (holomorphically) trivial on U . Let e , . . . , e n be an orthonormal basis for T ∗ U .For f ∈ T ∗ U , let c ( f ) ∈ End ( T ∗ , • U ) be the natural Clifford action of f (see (4.2) or [6]). Asusual, for every strictly increasing multi-index J = ( j , . . . , j q ) we set | J | := q , e J := e j ∧ · · · e j q and c ( e J ) = c ( e j ) · · · c ( e j q ) . For T ∈ End ( T ∗ , • U ) , we can always write T = X ′| J |≤ n c ( e J ) T J ( T J ∈ C ),where X ′ denotes the summation over strictly increasing multiindices. For k ≤ n , we put(5.26) [ T ] k := X ′| J | = k T J e J ( ∈ C T ∗ k U ) . and a similar expression for [ T ] (without the subscript k ). We identity T with [ T ] without the dangerof confusion. We say that ord T ≤ k if T J = 0 , for all | J | > k , and ord T = k if ord T ≤ k and [ T ] k = 0 .A crucial result for our need here is an identity in Berenzin integral (see [6, Proposition 3.21,Definition 3.4 and (1.28)]) which asserts that if ord T < n then STr T = 0 (see (1.13) for thedefinition of supertrace there) and(5.27) STr T = ( − i ) n STr T J c ( e J ) , J = (1 , , . . . , n ) . Recall the identification T ∗ , • x U ∼ = T ∗ , • U just mentioned above the theorem, so that a smoothfunction F ( x ) ∈ ( T ∗ , • U ) ⊠ ( T ∗ , • U ) ∗ | (0 ,x ) is identified with a function x → F ( x ) ∈ End ( T ∗ , • U ) ,giving a Taylor expansion F ( x ) = X α ∈ N n , | α |≤ P x α F α + O ( | x | P +1 ) , F α ∈ End ( T ∗ , • U ) . We are ready to apply Getzler’s rescaling technique to off-diagonal estimates. Consider A B ( t, x, y ) ≡ A B ( t, z, w ) := A B, + ( t, z, w ) ⊕ A B, − ( t, z, w ) (cf. Proposition 5.6) and let χ ∈ C ∞ ( U ) with χ = 1 near z = 0 . Let(5.28) r ( u, t, x ) := n X k =1 u − k + n [ χ ( √ ux ) A B ( ut, , √ ux )] k . EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 55 Note that A B is actually identified with [ A B ] similar to the case of T above, so that the k -form part( k > n ) of (5.28) makes sense.It is well-known that (see [6]) lim u → r ( u, t, x ) = g ( t, x ) ∈ C ∞ ( R + × C n , C T ∗• C n ) in C ∞ -topologylocally uniformly ( C T ∗• C n = ⊕ k =2 nk =0 Λ k C T ∗ C n ). In particular, lim u → r ( u, , x ) n = g (1 , x ) n in C ∞ -topology locally uniformly, for their n -form parts.Let b s ( z, w ) := b + s ( z, w ) ⊕ b − s ( z, w ) , s = n, n − , . . . (cf. (5.19)). One sees(5.29) r ( u, , x ) n = e − h +(0 , √ ux ) u χ ( √ ux ) (cid:16) u − n [ b n (0 , √ ux )] n + u − n +1 [ b n − (0 , √ ux )] n + · · · (cid:17) . Since lim u → e − h +(0 , √ ux ) u converges to a smooth function in C ∞ -topology locally uniformly on C n (see(5.19) in Proposition 5.6), we deduce that(5.30) lim u → (cid:16) u − n [ b n (0 , √ ux )] n + u − n +1 [ b n − (0 , √ ux )] n + · · · (cid:17) = ˆ g ( x ) ∈ C ∞ ( C n , C T ∗• C n ) in C ∞ -topology locally uniformly. Fix P ≫ . Write(5.31) ˆ g ( x ) = X α ∈ N n , | α |≤ P ˆ g α x α + O ( | x | P +1 ) . and for each s = n, n − , . . . ,(5.32) b s (0 , x ) = X α ∈ N n , | α |≤ P b s,α x α + O ( | x | P +1 ) . Hence(5.33) [ b s (0 , √ ux )] n = X α ∈ N n , | α |≤ P u | α | [ b s,α ] n x α + u P +12 O ( | x | P +1 ) , s = n, n − , . . . , and from (5.30), (5.31) and (5.33) it follows that for every α ∈ N n (5.34) lim u → (cid:16) u − n + | α | [ b n,α ] n + u − n +1+ | α | [ b n − ,α ] n + · · · (cid:17) = ˆ g α . With (5.34) we conclude(5.35) [ b n,α ] n = 0 , ∀ | α | < n. Combining (5.35) and (5.32), we see Tr b + n (0 , w ) − Tr b − n (0 , w ) = O ( | w | n ) . We can repeat the method above for the second leading term, and deduce similarly Tr b + s (0 , w ) − Tr b − s (0 , w ) = O ( | w | s ) , s = n − , n − , . . . , . The theorem follows. (cid:3) Heat kernels of the modified Kohn Laplacians ( Spin c Kohn Laplacians). Based on Proposi-tion 5.1, one is tempted to patch up the local heat kernels of the modified Kodaira Laplacian con-structed in Propostion 5.6 to form a global heat kernel for the modified Kohn Laplacian. This is noproblem in the globally free case (of the S action). In the locally free case, however, some delicatepoints arise as the relation of the two Laplacians given in the above proposition is, by nature, a localproperty, whereas the heat kernels are global objects. See discussions after proof of Proposition 5.1and of Theorem 5.14 with (5.54) for more.As remarked in Subsection 1.7.3, if we use the adjoint version of the original heat equation, itbecomes more effective to go over the desired process of patching up. It is worth noting that animportant role, mostly unseen traditionally, is played by the projection Q ± m in our situation.Assume X = D S D S · · · S D N with each B j := ( D j , ( z, θ ) , ϕ j ) a BRT trivialization. A slightlymore complicated set up is as follows. Write for each j , D j = U j × ] − δ j , e δ j [ ⊂ C n × R , δ j > , EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 56 e δ j > , U j = { z ∈ C n ; | z | < γ j } . Put ˆ D j = ˆ U j × ] − δ j , e δ j [ , ˆ U j = (cid:8) z ∈ C n ; | z | < γ j (cid:9) . We suppose X = ˆ D S ˆ D S · · · S ˆ D N .Here are some cut-off functions; the choice is adapted to BRT trivializations.i) χ j ( x ) ∈ C ∞ ( ˆ D j ) with P Nj =1 χ j = 1 on X . Put A j = n z ∈ ˆ U j ; there is a θ ∈ ] − δ j , e δ j [ such that χ j ( z, θ ) = 0 o . ii) τ j ( z ) ∈ C ∞ ( ˆ U j ) with τ j ≡ on some neighborhood W j of A j .iii) σ j ∈ C ∞ (] − δ j , e δ j [) with R e δ j / − δ j / σ j ( θ ) dθ = 1 .iv) ˆ σ j ∈ C ∞ (] − δ j , e δ j [) such that ˆ σ j = 1 on some neighbourhood of Supp σ j and ˆ σ j ( θ ) = 1 if ( z, θ ) ∈ Supp χ j .Write x = ( z, θ ) , y = ( w, η ) ∈ C n × R . We are going to lift many objects in the preceding subsectiondefined on U j to the ones defined on ˆ D j via the above cut-off functions.Let A B j , + ( t, z, w ) , K B j , + , h j, + ( z, w ) and b + j,s ( s = n, n − , . . . ) be as in Proposition 5.6 and (5.19).Slightly tediously, we put H j ( t, x, y ) = H j, + ( t, x, y ) = χ j ( x ) e − mϕ j ( z ) − imθ A B j , + ( t, z, w ) e mϕ j ( w )+ imη τ j ( w ) σ j ( η ) G j ( t, x, y ) = G j, + ( t, x, y ) = χ j ( x ) e − mϕ j ( z ) − imθ A B j , + ( t, z, w ) e mϕ j ( w )+ imη τ j ( w ) . (5.36)and (the last two equations are from (5.19)) ˆ K j, + ( t, x, y ) = ˆ K j ( t, x, y ) = χ j ( x ) e − mϕ j ( z ) − imθ K B j , + ( t, z, w ) e mϕ j ( w )+ imη τ j ( w ) σ j ( η )ˆ h j, + ( x, y ) = ˆ σ j ( θ ) h j, + ( z, w )ˆ σ j ( η ) ∈ C ∞ ( D j ) , x = ( z, θ ) , y = ( w, η )ˆ b + j,s ( x, y ) = χ j ( x ) e − mϕ j ( z ) − imθ b + j,s ( z, w ) e mϕ j ( w )+ imη τ j ( w ) σ j ( η ) , s = n, n − , . . . ˆ β + j,s ( x, y ) = χ j ( x ) e − mϕ j ( z ) − imθ b + j,s ( z, w ) e mϕ j ( w )+ imη τ j ( w ) , s = n, n − , . . .A B j , + ( t, z, w ) = e − h +( z,w ) t K B j , + ( t, z, w ) K B j , + ( t, z, w ) ∼ t − n b + j,n ( z, w ) + t − n +1 b + j,n − ( z, w ) + · · · + b + j, ( z, w ) + tb + j, − ( z, w ) + · · · as t → + . (5.37)Remark that these expressions, apart from cut-off functions, are mainly motivated by the formulas(5.5), (5.6) of Proposition 5.1. Let H j ( t ) be the continuous operator associated with H j ( t, x, y ) , forwhich we put down the expression for later use (cf. (5.42) and (5.44)) H j ( t ) : Ω , + ( X, E ) → Ω , + ( X, E ) ,u → Z χ j ( x ) e − mϕ j ( z ) − imθ A B j , + ( t, z, w ) e mϕ j ( w )+ imη τ j ( w ) σ j ( η ) u ( y ) dv X ( y ) . (5.38)Consider the patched up kernel (recall Q m is the projection on the m -th Fourier component, cf.(4.16))(5.39) Γ( t ) := N X j =1 H j ( t ) ◦ Q m : Ω , + ( X, E ) → Ω , + ( X, E ) and let Γ( t, x, y ) ∈ C ∞ ( R + × X × X, ( T ∗ , + X ⊗ E ) ⊠ ( T ∗ , + X ⊗ E ) ∗ ) be the distribution kernel of Γ( t ) . EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 57 For an explicit expression, one sees (using Q m of (4.17)) that Γ( t, x, y ) = 12 π N X j =1 Z π − π H j ( t, x, e − iu ◦ y ) e − imu du = 12 π N X j =1 Z π − π e − ˆ hj, +( x,e − iu ◦ y ) t ˆ K j ( t, x, e − iu ◦ y ) e − imu du ∼ t − n a + n ( t, x, y ) + t − n +1 a + n − ( t, x, y ) + · · · as t → + , (5.40)where we have written a + s ( t, x, y ) (= a + s,m ( t, x, y )) (ˆ b + j,s = ˆ b + j,s,m )= 12 π N X j =1 Z π − π e − ˆ hj, +( x,e − iu ◦ y ) t ˆ b + j,s ( x, e − iu ◦ y ) e − imu du, s = n, n − , n − , . . . . (5.41)For the initial condition of Γ( t, x, y ) , one has the following. Lemma 5.10. lim t → + Γ( t ) u = Q m u ( on D ′ ( X, T ∗ , + X ⊗ E )) for u ∈ Ω , + ( X, E ) .Proof. For u ∈ Ω , + ( X, E ) , Q m u ∈ Ω , + m ( X, E ) , Q m u | D j can be expressed as e − imη v j ( w ) for some v j ( w ) ∈ Ω , + ( U j , E ) . With (5.38) we find (note A B j , + ( t ) = I as t → and dv D j = dv U j dη by (2.5)) lim t → + H j ( t ) Q m u = lim t → + Z χ j ( x ) e − mϕ j ( z ) − imθ A B j , + ( t, z, w ) e mϕ j ( w )+ imη τ j ( w ) σ j ( η ) e − imη v j ( w ) dv U j ( w ) dη = lim t → + Z χ j ( x ) e − mϕ j ( z ) − imθ A B j , + ( t, z, w ) e mϕ j ( w ) τ j ( w ) v j ( w ) dv U j ( w )= χ j ( x ) e − mϕ j ( z ) − imθ e mϕ j ( z ) τ j ( z ) v j ( z )= χ j e − imθ v j = χ j Q m u. (5.42)With the above, the lemma follows from (5.39) and P j χ j = 1 . (cid:3) Γ( t ) satisfies an adjoint type heat equation asymptotically in the following sense (cf. [23, p. 96]). Lemma 5.11. We consider e (cid:3) + b,m ◦ Q m still denoted by e (cid:3) + b,m . Γ( t, x, y ) satisfies Γ ′ ( t ) u + Γ( t ) e (cid:3) + b,m u = R ( t ) u, ∀ u ∈ Ω , + ( X, E ) , where R ( t ) : Ω , + ( X, E ) → Ω , + ( X, E ) is the continuous operator with distribution kernel R ( t, x, y ) ( ∈ C ∞ ( R + × X × X, ( T ∗ , + X ⊗ E ) ⊠ ( T ∗ , + X ⊗ E ) ∗ ) ) which satisfies the following. For every ℓ ∈ N ,there exists an ε > , C ℓ > independent of t such that (5.43) k R ( t, x, y ) k C ℓ ( X × X ) ≤ C ℓ e − ε t , ∀ t ∈ R + . EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 58 Proof. As in the preceding lemma let u ∈ Ω , + m ( X, E ) and write u = e − imη v j ( w ) for some v j ( w ) ∈ Ω , + ( U j , E ) on D j . By this, (5.5), (5.16) and (5.38) it is a bit tedious but straightforward to compute H ′ j ( t ) u + H j ( t ) e (cid:3) + b,m u = Z χ j ( x ) e − mϕ j ( z ) − imθ A ′ B j , + ( t, z, w ) e mϕ j ( w )+ imη τ j ( w ) σ j ( η ) u ( y ) dv X ( y )+ Z χ j ( x ) e − mϕ j ( z ) − imθ A B j , + ( t, z, w ) e mϕ j ( w )+ imη τ j ( w ) σ j ( η )( e (cid:3) + b,m u )( y ) dv X ( y )= Z χ j ( x ) e − mϕ j ( z ) − imθ A ′ B j , + ( t, z, w ) τ j ( w ) e mϕ j ( w ) v j ( w ) dv U j ( w )+ Z χ j ( x ) e − mϕ j ( z ) − imθ A B j , + ( t, z, w ) τ j ( w )( e (cid:3) + B j ,m ( e mϕ j v j ))( w ) dv U j ( w )= Z χ j ( x ) e − mϕ j ( z ) − imθ A ′ B j , + ( t, z, w ) τ j ( w ) e mϕ j ( w ) v j ( w ) dv U j ( w )+ Z χ j ( x ) e − mϕ j ( z ) − imθ A B j , + ( t, z, w )( e (cid:3) + B j ,m ( τ j e mϕ j v j ))( w ) dv U j ( w )+ Z χ j ( x ) S j ( t, x, w ) v j ( w ) dv U j ( w )= Z χ j ( x ) S j ( t, x, w ) v j ( w ) dv U j ( w ) = Z χ j ( x ) S j ( t, x, w ) e imη σ j ( η ) u ( y ) dv X ( y ) , (5.44)for some S j ( t, x, w ) ∈ C ∞ ( R + × D j × U j , ( T ∗ , + X ⊗ E ) ⊠ ( T ∗ , + X ⊗ E ) ∗ ) .Note τ j ( z ) = 1 for ( z, θ ) in some small neighborhood of Supp χ j . One sees that S j ( t, x, w ) = 0 if ( x, w ) is in some small neighborhood of ( z, z ) . Hence by using (5.17) for (5.44) (on | z − w | away fromzero), we conclude that for every ℓ ∈ N , there is an ε > independent of t such that(5.45) k S j ( t, x, w ) k C ℓ ( X × X ) ≤ C ℓ e − εt , ∀ t ∈ R + . Put e R ( t, x, y ) := P Nj =1 χ j S j ( t, x, w ) e imη σ j ( η ) ( ∈ C ∞ ( R + × X × X, ( T ∗ , + X ⊗ E ) ⊠ ( T ∗ , + X ⊗ E ) ∗ )) and set R ( t, x, y ) = 12 π N X j =1 Z π − π e R ( t, x, e − iu ◦ y ) e − imu du. Let R ( t ) : Ω , + ( X, E ) → Ω , + ( X, E ) be the continuous operator with distribution kernel R ( t, x, y ) .Note that R ( t ) = R ( t ) ◦ Q m (cf. (5.40) and (5.39)). By (5.45) R ( t, x, y ) satisfies (5.43) and by (5.44)one sees Γ ′ ( t ) u + Γ( t ) e (cid:3) + b,m u = R ( t ) u , ∀ u ∈ Ω , + ( X, E ) . The lemma follows. (cid:3) To get back to the original heat equation from its adjoint version, it suffices to take the adjoints Γ ∗ ( t ) of Γ( t ) and R ∗ ( t ) of R ( t ) (with respect to ( · | · ) E ) because e (cid:3) + b,m is self-adjoint. Hence combiningLemma 5.10 and Lemma 5.11 one obtains the following (asymptotic) heat kernel. Theorem 5.12. With the notations above, we have lim t → + Γ ∗ ( t ) u = Q m u on D ′ ( X, T ∗ , + X ⊗ E ) for u ∈ Ω , + ( X, T ∗ , + X ⊗ E ) and ( e (cid:3) + b,m ◦ Q m still denoted by e (cid:3) + b,m below, which is self-adjoint) ∂ Γ ∗ ( t ) ∂t u + e (cid:3) + b,m Γ ∗ ( t ) u = R ∗ ( t ) u, ∀ u ∈ Ω , + ( X, E ) where R ∗ ( t ) is the continuous operator with the distribution kernel R ∗ ( t, x, y ) satisfying the estimatesimilar to Lemma 5.11. EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 59 Based on the above theorem, one way to solving our heat equation resorts to the method of suc-cessive approximation . This part of reasoning is basically standard. But because of the important roleplayed by Q + m in the final result (cf. McKean-Singer (II) in Corollary 5.15), for the convenience of thereader we sketch some details and refer the full details to, e.g. [6, Section 2.4].To start with, suppose A ( t ) , B ( t ) and C ( t ) : Ω , + ( X, E ) → Ω , + ( X, E ) are continuous operatorswith distribution kernels A ( t, x, y ) , B ( t, x, y ) and C ( t, x, y ) ( ∈ C ∞ ( R + × X × X, ( T ∗ , + X ⊗ E ) ⊠ ( T ∗ , + X ⊗ E ) ∗ )) . Define the (continuous) operator ( A♯B )( t ) : Ω , + ( X, E ) → Ω , + ( X, E ) with distri-bution kernel ( A♯B )( t, x, y ) := Z t Z X A ( t − s, x, z ) B ( s, z, y ) dv X ( z ) ds ( ∈ C ∞ ( R + × X × X, ( T ∗ , + X ⊗ E ) ⊠ ( T ∗ , + X ⊗ E ) ∗ )) . (5.46)It is standard that (( A♯B ) ♯C )( t ) = ( A♯ ( B♯C ))( t ) , denoted in common by ( A♯B♯C )( t ) . (The generalization to more operators is similar.)The method of successive approximation results in a solution (which is actually unique by Theo-rem 5.14 below) to our heat equation, as follows. Proposition 5.13. i) (Existence) Fix ℓ ∈ N , ℓ ≥ . There is an ǫ > such that the sequence (5.47) Λ( t ) := Γ ∗ ( t ) − (Γ ∗ ♯R ∗ )( t ) + (Γ ∗ ♯R ∗ ♯R ∗ )( t ) − · · · converges in C ℓ ((0 , ǫ ) × X × X, ( T ∗ , + X ⊗ E ) ⊠ ( T ∗ , + X ⊗ E ) ∗ ) and Λ( t ) : Ω , + ( X, E ) → C ℓ ( X, T ∗ , + X ⊗ E ) \ L , + m ( X, E ) , lim t → + Λ( t ) u = Q m u on D ′ ( X, T ∗ , + X ⊗ E ) , ∀ u ∈ Ω , + ( X, E ) , Λ ′ ( t ) u + e (cid:3) + b,m Λ( t ) u = 0 , ∀ u ∈ Ω , + ( X, E ) . (5.48) ii) (Approximation) Write Λ( t, x, y ) , ( ∈ C ℓ ((0 , ǫ ) × X × X, ( T ∗ , + X ⊗ E ) ⊠ ( T ∗ , + X ⊗ E ) ∗ )) for thedistribution kernel of Λ( t ) . Then there exists an ǫ > independent of t such that (5.49) k Λ( t, x, y ) − Γ ∗ ( t, x, y ) k C ℓ ( X × X ) ≤ e − ǫ t , ∀ t ∈ (0 , ǫ ) . With e (cid:3) − b,m in place of e (cid:3) + b,m in (5.48) , the corresponding statements for e (cid:3) − b,m (with Λ − , Γ − etc.) holdtrue as well.Proof. We sketche a proof of ii) and comment on i). For notational convenience we set Z = R ∗ , Z = R ∗ ♯R ∗ , Z = R ∗ ♯R ∗ ♯R ∗ etc. as defined in (5.46) with || · || ℓ as the C ℓ -norm on X × X . By using(5.43) (for R ∗ ) one sees that there are > δ , δ > such that for all t ∈ (0 , δ ) ,(5.50) k Z k ℓ ≤ e − δ t , (cid:13)(cid:13) Z (cid:13)(cid:13) ℓ ≤ e − δ t , · · · . Similarly from the estimate of Γ ∗ ( t ) (see (5.17)) with the above (5.50) we conclude that for all t ∈ (0 , δ ) , k Γ ∗ ♯Z k ℓ ≤ C e − δ t , (cid:13)(cid:13) Γ ∗ ♯Z (cid:13)(cid:13) ℓ ≤ C e − δ t , · · · (5.51)where C > is some constant. Hence the sequence (5.47) converges (in C ℓ ((0 , ǫ ) × X × X, ( T ∗ , + X ⊗ E ) ⊠ ( T ∗ , + X ⊗ E ) ∗ ) ) and (5.49) holds.It takes slightly more work to verify (5.48) in i). Let q k ( t, x, y ) be the ( k + 1) -th term in (5.47). Oneverifies directly by computation of the convolution that(5.52) ∂ t q k ( t, x, y ) + e (cid:3) + b,m q k ( t, x, y ) = Z k ( t, x, y ) + Z k +1 ( t, x, y ) EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 60 (cf. [6, (2) of Lemma 2.22]). Since Λ( t, x, y ) is the alternating sum of these q k , by the good estimates(5.50) and (5.51), one interchanges the order of the action of ( ∂ t + e (cid:3) + b,m ) on Λ( t, x, y ) with thesummation. By telescoping with (5.52), one finds that the heat equation (5.48) of i) is satisfied (cf.[6, Theorem 2.23]). (cid:3) The uniqueness part of the above theorem is included in the following. (Note e − t e (cid:3) + b,m ( x, y ) is as in(4.15).) Theorem 5.14. i) (Uniqueness) We have e − t e (cid:3) + b,m ( x, y ) = Λ( t, x, y ) ( ∈ C ℓ ((0 , ǫ ) × X × X, ( T ∗ , + X ⊗ E ) ⊠ ( T ∗ , + X ⊗ E ) ∗ )) . Hence by (5.49) , for every ℓ ∈ N there exist ǫ > and ǫ > (independent of t )such that (5.53) (cid:13)(cid:13)(cid:13) e − t e (cid:3) + b,m ( x, y ) − Γ( t, x, y ) (cid:13)(cid:13)(cid:13) C ℓ ( X × X ) ≤ e − ǫ t , ∀ t ∈ (0 , ǫ ) . As a consequence e − t e (cid:3) + b,m ( x, y ) and Γ( t, x, y ) are the same in the sense of asymptotic expansion (as definedin Definition 5.4).ii) (Asymptotic expansion) More explicitly one has (cf. (5.40) ) e − t e (cid:3) + b,m ( x, y ) ∼ t − n a + n ( t, x, y ) + t − n +1 a + n − ( t, x, y ) + · · · + a +0 ( t, x, y ) + ta + − ( t, x, y ) + · · · as t → + ,a + s ( t, x, y ) (= a + s,m ( t, x, y )) = 12 π N X j =1 Z π − π e − ˆ hj, +( x,e − iu ◦ y ) t ˆ b + j,s ( x, e − iu ◦ y ) e − imu du ( ∈ C ∞ ( R + × X × X, ( T ∗ , + X ⊗ E ) ⊠ ( T ∗ , + X ⊗ E ) ∗ )) , s = n, n − , n − , . . . . (5.54) Similar statements hold for the case of e − t e (cid:3) − b,m ( x, y ) as well.Proof. The argument for the uniqueness part is standard. To sketch it, there is the following trick (cf.[6, Lemma 2.16]) in which we shall use heat equations for both kernels (cf. (4.18) and (5.48)). For < t < ǫ ( ǫ as in Proposition 5.13) and f, g ∈ Ω , + ( X, E )0 = Z t ∂∂s (cid:16) ( Λ( t − s ) f | e − s e (cid:3) + b,m g ) E (cid:17) ds = ( Q m f | e − t e (cid:3) + b,m g ) E − ( Λ( t ) f | Q m g ) E = ( f | e − t e (cid:3) + b,m g ) E − ( Λ( t ) f | g ) E = ( e − t e (cid:3) + b,m f | g ) E − ( Λ( t ) f | g ) E , proving that e − t e (cid:3) + b,m ( x, y ) = Λ( t, x, y ) .The estimates (5.53) and (5.54) follow from (5.49) and (5.40) ( e − t e (cid:3) + b,m is self-adjoint). (cid:3) Remark that by Proposition 5.1 it was tempting to speculate that the heat kernel for (modified/ Spin c )Kohn Laplacian might be (at least asymptotically) the (local) heat kernel for (modified/ Spin c ) KodairaLaplacian. This is however too much to be true as suggested by the above Theorem 5.14 because theasymptotic expansion of (modified) Kohn Laplacian involves a nontrivial t -dependence in a s ( t, x, y ) (cf. Remark 1.6 and Remark 1.7).We are ready to establish a link between our index and the heat kernel density of (modified) KodairaLaplacian. For a + ℓ ( t, x, x ) in (5.54), define Tr a + ℓ ( t, x, x ) = P ds =1 h a + ℓ ( t, x, x ) e s ( x ) | e s ( x ) i E as usual,where { e s ( x ) } s an othonormal frame (of T ∗ , + x X ⊗ E x ). (Similar notation and definition apply to thecase of a − ℓ ( t, x, x ) . )To sum up from Corollary 4.8 and (5.54), there is a second form of McKean-Singer type formula forthe index in our case (cf. Corollary 4.8 for the first form). EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 61 Corollary 5.15. (McKean-Singer (II)) We have (5.55) n X j =0 ( − j dim H jb,m ( X, E ) = lim t → + Z X n X ℓ =0 t − ℓ (cid:16) Tr a + ℓ ( t, x, x ) − Tr a − ℓ ( t, x, x ) (cid:17) dv X ( x ) . By this result we are now reduced to computing a s ( t, x, x ) in the following section. Part II: Proofs of main theorems 6. P ROOFS OF T HEOREMS AND ˆ d (see (1.16) for the definition and Theorem 6.7 for its property). This functionnaturally appears when we compute a s ( t, x, x ) in the form of an integral (5.54). In the remaining partof this section we prove that this “distance function” is equivalent to the ordinary distance function atleast in the strongly pseudoconvex case (Theorem 6.7).Theorem 1.3 is proved in Theorem 6.1, Remark 6.2, Corollary 6.3 together with Theorem 5.14;Theorem 1.10 proved in Theorem 6.4 and in (6.15).In the same notations as before recall that X p ℓ = n x ∈ X ; the period of x is πp ℓ o , ≤ ℓ ≤ k with p | p ℓ (all ℓ ) and p = p . X p is open and dense in X . See the discussion preceding Theorem 1.3 formore detail.Let G j ( t, x, y ) be as in (5.36). (Notations set up in (5.36)–(5.41) will be useful in what follows.)By the construction of G j ( t, x, y ) , it is clear that π N X j =1 G j ( t, x, x ) ∼ t − n α + n ( x ) + t − n +1 α + n − ( x ) + · · · as t → + ,α + s ( x ) = 12 π N X j =1 ˆ β + j,s ( x, y ) | y = x ∈ C ∞ ( X, End ( T ∗ , + X ⊗ E )) , s = n, n − , . . . . (6.1) α + s ( x ) are independent of choice of BRT trivialization charts (in view of Remark 5.7). It is perhapsinstructive to think of these as the data of the asymptotic expansion associated with the “underly-ing Kodaira Laplacian” (cf. loc. cit. and Proposition 5.1) regardless of the existence of a genuine“underlying space”.Recall the asymptotic expansions of Γ( t, x, y ) and e − t e (cid:3) + b,m ( x, y ) (they coincide by Theorem 5.14), inwhich we have a + s ( t, x, y ) ( ∈ C ∞ ( R + × X × X, ( T ∗ , + X ⊗ E ) ⊠ ( T ∗ , + X ⊗ E ) ∗ ) ), s = n, n − , . . . , cf.(5.54) or (5.40). By the construction, Γ( t, x, y ) and a s ( t, x, y ) of (5.54) depend on the choice of BRTcharts. (The authors do not know whether there exists a canonical choice of a s ( t, x, y ) in this respect.)We are now ready to give a proof of the following. Theorem 6.1. (cf. Theorem 1.3) For every N ∈ N with N ≥ N ( n ) for some N ( n ) , there exist ε > , δ > and C N > such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X j =0 t − n + j a + n − j ( t, x, x ) − (cid:16) p r X s =1 e π ( s − pr mi (cid:17) N X j =0 t − n + j α + n − j ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C N (cid:16) t − n + N +1 + t − n e − ε d ( x,Xr sing )2 t (cid:17) , ∀ x ∈ X p r ( r = 1 , . . . , k ) , ∀ < t < δ. (6.2) Proof. For simplicity, we only prove Theorem 6.1 for r = 1 . The proof for r > is similar. EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 62 As in the beginning of Section 5.2, there are BRT trivializations B j := ( D j , ( z, θ ) , ϕ j ) , j = 1 , . . . , N .We write D j = U j × ] − δ j , e δ j [ , ˆ D j = ˆ U j × ] − δ j , e δ j ⊂ C n × R ) , with U j = { z ∈ C n ; | z | < γ j } , ˆ U j = { z ∈ C n ; | z | < γ j / } for some δ j > , e δ j > , γ j > .Assume X = ˆ D S · · · S ˆ D N . In the following we let δ j = e δ j = ζ (all j ), ζ satisfy (1.15) with | ζ | < πp .It is easily verified that there is an ˆ ε > such that ( d ( · , · ) the ordinary distance function on X ) ˆ ε d (( z , θ ) , ( z , θ )) ≤ | z − z | ≤ ε d (( z , θ ) , ( z , θ )) , ∀ ( z , θ ) , ( z , θ ) ∈ D j , ˆ ε d (( z , θ ) , ( z , θ )) ≤ h j, + ( z , z ) ≤ ε d (( z , θ ) , ( z , θ )) , ∀ ( z , θ ) , ( z , θ ) ∈ D j , (6.3)where h j, + ( z, w ) is as in (5.19).Recall the modified distance ˆ d which is defined in (1.16). We are going to compare ˆ d with (6.3).Fix x ∈ X p . Suppose x ∈ ˆ D j for some j = 1 , , . . . , N and also suppose x = ( z, on D j .Some crucial remarks are in order.i) For ≤ | u | ≤ ζ the action of e − iu on x is only moving along the “angle” direction (due to theassumption that a BRT trivialization D j is valid here), i.e. the z -coordinates of x and e − iu ◦ x are thesame.ii) Let a u ∈ [2 ζ, πp − ζ ] be given. Assume that the action by e − iu on x still belongs to ˆ D j witha coordinate e − iu ◦ x = (˜ z, ˜ η ) . Then it could happen that ˜ z = z because the orbit { e − iv ◦ x } for ζ ≤ v ≤ πp − ζ may partly lie outside of D j . We will show (6.4) below that indeed ˜ z = z in thiscase.Remark that the above ii) is basically the reason responsible for why the contribution of our distancefunction ˆ d enters, as seen shortly. The question about whether the condition e − iu ◦ x ∈ ˆ D j of ii) isvacuous or not will be discussed below (equivalent to whether J below is an empty set or not).We shall now formulate the above ii) more precisely. If x ∈ ˆ D j , we claim the following.Suppose e − iθ ◦ x = ( e z, e η ) also belongs to ˆ D j for some θ ∈ [2 ζ, πp − ζ ] . Then | z − e z | ≥ ˆ ε ˆ d ( x , X sing ) ( > . (6.4) proof of claim. By (˜ z, ˜ η ) ∈ ˆ D j one has e i ˜ η ◦ (˜ z, ˜ η ) = (˜ z, equivalently e − i ˜ η ◦ (˜ z, 0) = (˜ z, ˜ η ) (by theabove i) as | ˜ η | ≤ ζ here). One sees (by (6.3) and isometry of S action for the first inequality below) | e z − z | ≥ ˆ ε d ( e i ˜ η ◦ (˜ z, ˜ η ) , e i ˜ η ◦ ( z, ˜ η )) = ˆ ε d ( e i ˜ η ◦ ( e − iθ ◦ x ) , x ) ≥ ˆ ε inf (cid:26) d ( e − iu ◦ e − iθ ◦ x , x ); | u | ≤ ζ (cid:27) ≥ ˆ ε inf (cid:26) d ( e − i ˆ θ ◦ x , x ); ζ ≤ ˆ θ ≤ πp − ζ (cid:27) = ˆ ε ˆ d ( x , X sing ) (6.5)(see (1.16) for the definition of ˆ d ), as claimed. (cid:3) Remark that a sharp result in this direction (6.4) is proved in Lemma 7.6.We continue with the proof of the theorem. We need to estimate Γ( t ) = P Nj =1 H j ( t ) ◦ Q m for thefirst summation to the left of (6.2). By definition (see (5.40)) this is in turn to estimate(6.6) π Z π − π H j ( t, x , e − iu ◦ x ) e − imu du and sum over j = 1 , . . . , N . EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 63 We first assume that in (6.6), x = ( z, in ˆ D j and x ˆ D k for any other k = j .To work on (6.6) we shall divide [ − π, π ] in (6.6) into two types.The first type is to estimate π R ζ − ζ H j ( t, x , e − iu ◦ x ) e − imu du . Note if u ∈ [ − ζ, ζ ] and e − iu ◦ x =( z u , θ u ) , then ( z u , θ u ) = ( z, u ) (by i) above (6.4)), i.e. e − iu ◦ ( z, 0) = ( z, u ) . Hence, by (5.36) thefactor e imη in H j is going to be e imu and this is cancelling off the term e − imu in the integral (6.6). By(5.19), h + ( z, z u ) = h + ( z, z ) = 0 (also by (6.3)) and the factor e − h + t of A B j , + in H j of (5.36) becomes . Finally we note for σ j in H j of (5.36), R I σ j ( θ ) dθ = 1 , I = [ − ζ , ζ ] .To sum up, by (5.36) and (5.19) one obtains the following ( x ˆ D k for k = j ) π Z ζ − ζ H j ( t, x , e − iu ◦ x ) e − imu du ∼ (cid:16) t − n α + n ( x ) + t − ( n − α + n − ( x ) + · · · (cid:17) as t → + , (6.7)where α + s ( x ) , s = n, n − , . . . , are as in (6.1).For the second type suppose u ∈ [2 ζ, πp − ζ ] . Note the action by e − iu on x may change the z coordinate of x by ii) above (6.4). We let J be the subset of those u ∈ [2 ζ, πp − ζ ] ≡ E that e − iu ◦ x = ( z u , θ u ) belongs to ˆ D j (then z u = z by (6.4)), and J ′ be the complement of J in E . Onefinds, for some ε > , δ > and C > (independent of j , x ), that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π Z u ∈ [2 ζ, πp − ζ ] H j ( t, x , e − iu ◦ x ) e − imu du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ π Z u ∈ J (cid:12)(cid:12) H j ( t, x , e − iu ◦ x ) e − imu (cid:12)(cid:12) du + 12 π Z u ∈ J ′ (cid:12)(cid:12) H j ( t, x , e − iu ◦ x ) e − imu (cid:12)(cid:12) du ≤ C t − n e − ε d ( x ,X sing )2 t , ∀ < t < δ (6.8)where the integral over J ′ vanishes because the cut-off function σ j in H j of (5.36) gives σ j ( η ( y )) = 0 for y = e − iu ◦ x ˆ D j as σ j = 0 outside ˆ D j (see lines above (5.36)), and the second inequality arisesfrom applying (6.3) and (6.4) to h + ( z, z u ) in H j (see (5.36) and (5.19)).Is J an empty set? We remark that the top term in (6.8) is in general nonzero (by combining (6.7)and Remark 1.7 for p = 1 ). Hence J = ∅ in general. There is a geometrical way to see the claimthat for some open subset V of X , if x ∈ V , then J = ∅ . For simplicity assume X = X S X , i.e. p = 1 and p = 2 . Choose y ∈ X . Let g = e − i πp ∈ S . Fix a neighborhood U ⊂ ˆ D j of y in X . Since g ◦ y = y , by continuity argument there are neighborhoods N , N of y , g in X , S respectively suchthat the action h ◦ x ∈ U if ( h, x ) ∈ N × N . Choose N ⊂ ˆ D j , N small and set V ≡ N \ X . Itfollows that for these x ∈ V , J = ∅ since N ⊂ J . This result also accounts for the necessity of theremark ii) above (6.4) and hence that of a certain extra contribution (e.g. ˆ d ) in estimates (6.8).Suppose p (= p ) = 1 . Then (6.7) and (6.8), Definition 5.4 for ∼ and Remark 5.5 (by noting dim X = 2 n + 1 , dim U j = 2 n , M ( m ) = n + 1 , m = 4 n = 2 β , M ( m, ℓ ) = n for ℓ = 0 ) immediatelylead to (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ( t, x , x ) − N X j =0 t − n + j α + n − j ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C N (cid:16) t − n + N + C t − n e − ε d ( x ,X sing )2 t (cid:17) , N ≥ N ( n ) , ∀ < t < δ. (6.9)Now adding t − n + N +1 α + n − N − to (6.9) and substracting it we improve t − n + N by t − n + N +1 . Hencethe estimate (6.2) of the theorem for p = 1 . EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 64 Suppose p > . Then one has the extra p − sectors in [ − π, π ] (obtained by shifting the above firstsector s = 1 by a common ( s − πp ):(6.10) [( s − 1) 2 πp − ζ, ( s − 1) 2 πp + 2 ζ ] , [( s − 1) 2 πp + 2 ζ, s πp − ζ ] , s = 1 , . . . , p ( s = p + 1 identified with s = 1 ) over which the integrals correspond to types I (6.7) and II (6.8)respectively. One may check without difficulty that the version of the claim (6.4) adapted to thesesectors holds true as well. On each of these sectors, a simple (linear) change of variable for u , whichis to bring the intervals of the integration on these sectors back to those in (6.7) and (6.8), producesthe extra numerical factor in sum (by e − imu du in (6.6)) : p P s =1 e π ( s − p mi as expressed in (6.2).Finally, note that we have assumed x = ( z, ∈ ˆ D j . In this case the above argument appears symmetrical in writing. This (assumption) is however not essential. Since we shall also adopt a similarassumption in Section 7, we give an outline about the asymmetrical way (i.e. x = ( z, θ ) , θ = 0 ) of theargument. By going over the same process, one sees the following. i) If x = ( z, v ) , with < v ≤ ζ ,the intervals in (6.7), (6.8) shall be replaced by [ − ζ − v, ζ − v ] , [2 ζ − v, πp − ζ − v ] (thoughtof as translated by a common − v ) with the new integrals denoted by (6.7)’, (6.8)’, respectively; ii) [ − ζ − v, ζ − v ] ⊇ [ − ζ, ζ ] hence R σ ( u ) du is still in (6.7)’ ; iii) In the proof of claim (6.5), e i ˜ η should bereplaced by e iγ with γ = ˜ η − v , (˜ z, by (˜ z, v ) and θ ∈ [2 ζ, πp − ζ ] by θ ∈ [2 ζ − v, πp − ζ − v ] throughout(6.4) and (6.5). One can check that the the reasoning in (6.5) remains basically unchanged, and theconclusion of (6.5) holds true as well in this modified case; iv) By the preceding ii) and iii), the resultscorresponding to (6.7)’ and (6.8)’ hold true. Hence the asymmetrical way follows.We have also assumed x ˆ D k for k = j . This condition is unimportant if we take the precedingasymmetrical way into account (for x ∈ ˆ D k , k = j in the general case), and note that there is ahidden partition of unity in { H j } j =1 ,...,N .For an alternative to the above, it is to use the kernel e − t e (cid:3) + b,m ( x, y ) in place of Γ( t, x, y ) and a + s ( t, x, y ) . An advantage is that e − t e (cid:3) + b,m ( x, y ) is independent of BRT charts, so that for a given point x we can take a covering of X by convenient BRT charts for the previous special conditions to be satis-fied (e.g. x = ( z, , x ∈ ˆ D j for exactly one j etc.). By the asymptotic property between e − t e (cid:3) + b,m ( x, y ) and Γ( t, x, y ) (5.53), this also leads to Theorem 6.1. (cid:3) Remark . For the relation between a + s ( t, x, y ) | y = x and α s ( x ) as stated in (1.18) of Theorem 1.3, themethod of the above proof works. By using (setting y = x below)(6.11) a + s ( t, x, y ) = 12 π N X j =1 Z π − π e − ˆ hj, +( x,e − iu ◦ y ) t ˆ b + j,s ( x, e − iu ◦ y ) e − imu du (see (5.54)) with the same reasoning as (6.7), (6.8) and (6.10), one obtains (1.18) for P ℓ = id . For ℓ > , (1.18) follows by noting(6.12) ∂ x e − x t = − t − ( x t ) / e − x t = O ( t − ) applied to (6.11) and by noting Remark 5.5 as in (6.9). Remark that if one extracts the correspondingcoefficients of t − s in (6.2) of Theorem 6.1 and uses the result (6.2), the estimate appears to be e − ε d ( x,X sing) t + O ( t ∞ ) which is slightly weaker than above (due to O ( t ∞ ) ).For similar estimates with regard to C l topology we have the following (cf. (6.12) and Remark 5.5). EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 65 Corollary 6.3. In the same notation as above, for any differential operator P ℓ : C ∞ ( X, T ∗ , + X ⊗ E ) → C ∞ ( X, T ∗ , + X ⊗ E ) of order ℓ ∈ N and every N ≥ N ( n ) for some N ( n ) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ℓ (cid:16) N X j =0 t − n + j a + n − j ( t, x, x ) − (cid:0) p r X s =1 e π ( s − pr mi (cid:1) N X j =0 t − n + j α + n − j ( x ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C N (cid:16) t − n + N +1 − ℓ + t − n − ℓ e − ε d ( x,Xr sing )2 t (cid:17) , ∀ < t < δ, ∀ x ∈ X p r (6.13) for some ε > , δ > and C N > independent of x . Note the singular behavior t − n (in the term to the rightmost of (6.2)). So the estimate (6.2) isnot directly applicable to the proof of our local index theorem. That is, computation involving a s cannot be automatically reduced to computation involving α s as soon as x ( ∈ X p ) approaches X sing .Intuitively t − n e − ε d ( x,X sing )2 t goes to a kind of Dirac delta function (along X sing ) as t → (apart froma factor of the form t β , some β > ). So after integrating (6.2) over X , a nonzero contribution due tothis term could appear or even blow up as t → . A more precise analysis along this line will be takenup in the study of trace integrals in Section 7.Fortunately, the abovementioned singular behavior can be removed ( t − n dropping out completely)after taking the supertrace , so that the index density for our need does exist. (However, as far as thefull kernel is concerned, a certain estimate such as that in Theorem 6.1 is unavoidable as discussed inRemark 1.8).We shall now take up this improvement on (6.2) under supertrace. We formulate it as follows,whose proof is heavily based on the off-diagonal estimate obtained in Theorem 5.9. Theorem 6.4. (cf. Theorem 1.10) With the notations above, for every N ∈ N , N ≥ N ( n ) for some N ( n ) , there exist ε > , δ > and C N > such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Tr e − t e (cid:3) + b,m ( x, x ) − Tr e − t e (cid:3) − b,m ( x, x ) − (cid:16) p r X s =1 e π ( s − pr mi (cid:17) N X j =0 t − n + j (cid:16) Tr α + n − j ( x ) − Tr α − n − j ( x ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C N (cid:16) t − n + N +1 + e − ε d ( x,Xr sing )2 t (cid:17) , ∀ < t < δ, ∀ x ∈ X p r , (6.14)The implication of Theorem 6.4 yields a link between the two identities arising from Corollary 5.15and Proposition 5.8 together with (5.20): lim t → + Z X n X ℓ =0 t − ℓ (cid:16) Tr a + ℓ ( t, x, x ) − Tr a − ℓ ( t, x, x ) (cid:17) dv X ( x ) = n X j =0 ( − j dim H jb,m ( X, E )lim t → + Z X n X ℓ =0 t − ℓ (cid:16) Tr α + ℓ ( x ) − Tr α − ℓ ( x ) (cid:17) dv X ( x )= 12 π Z X Td b ( ∇ T , X , T , X ) ∧ ch b ( ∇ E , E ) ∧ e − m dω π ∧ ω ( x ) . (6.15)It follows that the two in (6.15) are equal because in (6.14), e − ε d ( x,X sing )2 t ( ≤ → in L byLebesgue’s dominated convergence theorem as t → + on X p . We arrive now at an index theorem forour class of CR manifolds. EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 66 Corollary 6.5. (cf. Corollary 1.13) n X j =0 ( − j dim H jb,m ( X, E )= (cid:16) p X s =1 e π ( s − p mi (cid:17) π Z X h Td b ( ∇ T , X , T , X ) ∧ ch b ( ∇ E , E ) ∧ e − m dω π ∧ ω i n +1 ( x ) (6.16) where [ · · · ] | n +1 denotes the (2 n + 1) -form part. We turn now to the proof of Theorem 6.4. proof of Theorem 6.4. For simplicity, we only prove Theorem 6.4 for r = 1 . The proof for r > issimilar. Adopting the same notations as in the proof of Theorem 6.1 (e.g. B j , D j , ˆ D j · · · ), we shallfollow a similar line of thought as in Theorem 6.1.Fix x ∈ X p . As e − t e (cid:3) + b,m ( x, y ) and Γ( t, x, y ) are asymptotically the same (Theorem 5.14), we alsobreak the desired estimate at x = x into two types of integrals corresponding to (6.7) and (6.8).One integral is over I = [ − ζ, ζ ] and the other over I ′ , the complement of I in [ − π, π ] . The firsttype gives rise to the first term to the right of (6.14) almost the same way as (6.7).The key of this proof lies in the second type which corresponds to (6.8). It is estimated over I ′ , as in(6.17) below. (Here we rewrite H j in a convenient form, in terms of ˆ h j, + , ˆ K j, + of (5.37), reminiscentof an analogous relation A B, + = e − h + t K B, + in (5.19).)(6.17) π N X j =1 (cid:12)(cid:12)(cid:12)(cid:12)Z u ∈ I ′ e − ˆ hj, +( x ,e − iu ◦ x t (cid:16) Tr ˆ K j, + ( t, x , e − iu ◦ x ) − Tr ˆ K j, − ( t, x , e − iu ◦ x ) (cid:17) du (cid:12)(cid:12)(cid:12)(cid:12) . We shall now show that there exist ε > and C > (independent of x ) such that (6.17) isbounded above by(6.18) Ce − ε d ( x ,X sing )2 t for small t ∈ R + .To see this we first note that for k ≥ ,(6.19) e − ε x t (cid:0) x t (cid:1) k ≤ C k,ε e − ε x t for some constant C k,ε independent of x and t > . Write x = ( z, θ ) and e − iu ◦ x = ( z u , θ u ) in BRTcoordinates. Since ˆ h j, + ( x , e iu ◦ x ) is essentially h + ( z, z u ) ≈ | z − z u | , we have(6.20) e − ˆ h ( x ,eiu ◦ x t ≤ e − c | z − zu | t ≤ e − c ε d ( x ,X sing )2 t e − c | z − zu | t for some constant c > by using (6.4) for ˆ d . By using the off-diagonal estimate of Theorme 5.9 andby (5.19), (5.37) for linking b • with K • , one obtains the following estimate from (6.20) (cid:12)(cid:12)(cid:12)(cid:12) e − ˆ hj, +( x ,e − iu ◦ x t Tr ˆ K j, + ( t, x , e − iu ◦ x ) − Tr ˆ K j, − ( t, x , e − iu ◦ x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ e − c ε d ( x ,X sing )2 t n X k =0 constants · e − c | z − zu | t (cid:0) | z − z u | k t k + O ( t ) (cid:1) (6.21)for (6.17). Now one readily obtains the bound (6.18) from (6.21) and (6.19).Combining the above estimates for integrals of the first type and second type (6.17), we obtain(6.14) in the way similar to (6.9) (with t − n dropping out of t − n e − ε d t ). (cid:3) EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 67 In the remaining part of this section we give a geometric meaning for ˆ d ( x, X r sing ) (when X isstrongly pseudoconvex). To this aim it is useful to use another equivalent form of the function ˆ d , asfollows (without any pseudoconvexity condition on X ). Lemma 6.6. There exists a small constant ε > (satisfying (1.15) at least) with the following property.Fix an ε with < ε ≤ ε . For x ∈ X define another “distance function” ˆ d by (for a fixed ℓ ) ˆ d ( x, X ℓ − ) = inf (cid:26) d ( x, e − iθ ◦ x ); 2 πp ℓ − ε ≤ θ ≤ πp ℓ + ε (cid:27) ( X ℓ − = X p ℓ S X p ℓ +1 · · · ). Then ˆ d ( x, X ℓ − ) is equivalent to ˆ d ( x, X ℓ − ) . (Namely C ℓ,ε ˆ d ≤ ˆ d ≤ C ℓ,ε ˆ d for some constant C ℓ,ε independent of x ). We postpone the proof of the lemma until after Theorem 6.7.For technical reasons we impose a pseudoconvex condition on X in the following although the sameresult is expected to hold without this condition. Theorem 6.7. With the notations above, assume that X is strongly pseudoconvex. Then there is aconstant C ≥ such that C d ( x, X r sing ) ≤ ˆ d ( x, X r sing ) ≤ Cd ( x, X r sing ) , ∀ x ∈ X. Proof. For simplicity, we assume that X = X S X , i.e. p = 1 , p = 2 , so that X sing ≡ X = X ( r = 1) by definition. For the general case, the proof is essentially the same. By Lemma 6.6, forevery (small and fixed) ε > we have(6.22) ˆ d ( x, X sing ) ≈ inf n d ( x, e − iθ ◦ x ); π − ε ≤ θ ≤ π + ε o . Since X is strongly pseudoconvex, it is well-known that (see [40]) there exists a CR embedding: Φ : X → C N ,x → ( f ( x ) , . . . , f N ( x )) (6.23)with f j ∈ H b,m j ( X ) for some m j ∈ N ( j = 1 , . . . , N ).We assume that m , . . . , m s are odd numbers and m s +1 , . . . , m N are even numbers. By p = 1 and p = 2 one sees that (cf. (1.35))(6.24) x ∈ X sing if and only if f ( x ) = · · · = f s ( x ) = 0 so that(6.25) d ( x, X sing ) ≈ s X j =1 | f j ( x ) | , ∀ x ∈ X. Now, by using the embedding theorem (6.23) (together with (1.35)) we have(6.26) d ( x, e − iπ ◦ x ) ≈ N X j =1 (cid:12)(cid:12) f j ( x ) − f j ( e − iπ ◦ x ) (cid:12)(cid:12) = 4 s X j =1 | f j ( x ) | ≈ d ( x, X sing ) and hence for every π − ε ≤ θ ≤ π + ε ( ε > small)(6.27) d ( x, e − iθ ◦ x ) ≈ N X j =1 (cid:12)(cid:12)(cid:12) f j ( x ) − f j ( e − iθ ◦ x ) (cid:12)(cid:12)(cid:12) ≥ s X j =1 (cid:12)(cid:12)(cid:12) (1 − e − im j θ ) f j ( x ) (cid:12)(cid:12)(cid:12) ≈ s X j =1 | f j ( x ) | ≈ d ( x, X sing ) . By (6.27) we conclude that(6.28) inf n d ( x, e − iθ ◦ x ); π − ε ≤ θ ≤ π + ε o ≈ d ( x, X sing ) . Combining (6.22) and (6.28) we have proved the theorem. (cid:3) EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 68 We give now: proof of Lemma 6.6. In the following we write ˆ d ( x ) = ˆ d ( x, X ℓ − ) and ˆ d ( x ) = ˆ d ( x, X ℓ − ) for a fixed ℓ . For an illustration we assume X = X S X , i.e. p = 1 , p = 2 , and x ∈ X ( ˆ d = ˆ d = 0 for x ∈ X .) Write I for the complement of I ′ ≡ ] π − ε, π + ε [ in [ ζ, π − ζ ] ≡ K (where ζ satisfies (1.15)and ε > a small constant to be specified, cf. the line above (6.35)). By definition ˆ d ≥ ˆ d (= ˆ d ζ ) ( ˆ d is to take inf over I ′ while ˆ d over K , and I ′ ⊂ K ).It remains to see ˆ d ≤ C ˆ d for some C . Put f S ( x ) = inf θ ∈ S n d ( x, e − iθ ◦ x ) o for a set S . We claim that there exists a c , > c > ,(6.29) f I ( x ) ≥ c for all x ∈ X . Indeed for each x ∈ X and for any θ ∈ I = [ ζ, π − ε ] ∪ [ π + ε, π − ζ ] one sees x = e − iθ ◦ x .So (6.29) follows by a compactness argument. Let M ≥ be an upper bound of ˆ d . We claim(6.30) ˆ d ( x ) ≤ Mc ˆ d ( x ) , x ∈ X. Note ˆ d ( x ) = f K ( x ) = min { f I ( x ) , f I ′ ( x ) } and ˆ d = f I ′ . Suppose f K ( y ) < f I ( y ) . Then f K ( y ) = f I ′ ( y ) ,i.e. ˆ d ( y ) = ˆ d ( y ) and (6.30) holds for these y (as Mc > ). If f K ( y ) ≥ f I ( y ) (for some y ∈ X ), then f K ( y ) = f I ( y ) , giving ˆ d ( y ) ≥ c by (6.29). For these y , (6.30) still holds. In any case we have proved(6.30) for x ∈ X , hence for x ∈ X ( ˆ d = ˆ d = 0 at x ∈ X ).For another illustration, in the same notation as above except that say, X = X S X S X (i.e. p = 4 ). We are going to prove the lemma for the case ℓ = 2 (with x ∈ X , as ˆ d = ˆ d = 0 at x X for ℓ = 2 ).With the above I, I ′ and K , let J be the complement of J ′ ≡ ] π − ε, π + ε [ S ] π − ε, π + ε [ in I .It follows, similarly as (6.29), that there exists a c , > c > such that(6.31) f J ( x ) ≥ c , ∀ x ∈ X. Let { W α } α be the set of connected components of X . Each y ∈ X is a fixed point of the subgroup Z = { , e i π , e iπ , e i π } of S ; write λ i,α ( g ) for all the eigenvalues of the isotropy (and isometric) actionof g ∈ Z on T y X for y ∈ W α . All of them are independent of the choice of y ∈ W α . Let C M = max = g ∈ Z ,λ i,α ( g ) {| λ i,α ( g ) − |} > c m = min = g ∈ Z ,λ i,α ( g ) =1 {| λ i,α ( g ) − |} > . Let B = { x ∈ X ; ˆ d ( x ) ≥ ( C M c m + 1) Mc ˆ d ( x ) > } ( M ≥ as above). Clearly B ⊂ X (zero distancefor x ∈ X ∪ X ). We claim that(6.32) B ∩ X = ∅ . To see (6.32) suppose otherwise. Let y n ∈ B and y n → y ∈ X as n → ∞ . Observe that f K ( y n ) = f I ′ ( y n ) for all n because the equality ˆ d ( y n ) = ˆ d ( y n ) (note f K = ˆ d and f I ′ = ˆ d ) clearly contradicts thedefinition of B with y n ∈ B . By K = I ′ ∪ J ′ ∪ J , we are left with two possibilities for a y n i) f K ( y n ) = f J ′ ( y n )ii) f K ( y n ) = f J ( y n ) . (6.33)Suppose i). By examining the isotropy (and isometric) action of Z at y ∈ X , one sees that bothratios below(6.34) d ( y n , e ı π ◦ y n ) d ( y n , e ı π ◦ y n ) , d ( y n , e ı π ◦ y n ) d ( y n , e ı π ◦ y n ) EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 69 are bounded above by C M c m + as n ≫ . Since I ′ and J ′ are ε -neighborhoods around π and { π , π } respectively, by choosing a sufficiently small ε (say ε ≤ ε ) one sees from (6.34)(6.35) f I ′ ( y n ) f J ′ ( y n ) ≤ C M c m + 12 , n ≫ We claim that this contradicts y n ∈ B . Note f K = ˆ d and f I ′ = ˆ d so that the assmption i) f K ( y n ) = f J ′ ( y n ) amounts to ˆ d ( y n ) = f J ′ ( y n ) and (6.35) gives(6.36) ˆ d ( y n )ˆ d ( y n ) ≤ C M c m + 12 , n ≫ . By y n ∈ B , (6.36) contradicts the definition of B .Suppose ii) of (6.33). By (6.31), f J ( x ) ≥ c for all x ∈ X hence by f K = ˆ d and ii) of (6.33), oneobtains ˆ d ( y n ) ≥ c , giving ˆ d ( y n ) ≥ ( C M c m + 1) M by using y n ∈ B , which is absurd since ˆ d ≤ M byassumption. The claim (6.32) is proved by contradictions in i) and ii) of (6.33).Granting the claim (6.32) we have B ⊂ X ∪ X (which is open in X ). Since for θ ∈ I and x ∈ X ∪ X (in particular for x ∈ B ) x = e − iθ ◦ x , by compactness there exists a c , > c > satisfying (as in (6.29))(6.37) f I ( x ) ≥ c for all x ∈ B . One asserts that(6.38) ˆ d ( x ) ≤ ( C M c m + 1) Mc c ˆ d ( x ) , ∀ x ∈ B. The argument is similar. By f K = min { f I , f I ′ } , a) f K ( x ) = f I ′ ( x ) or b) f K ( x ) = f I ( x ) . a) If f K ( x ) = f I ′ ( x ) , then by f K = ˆ d and f I ′ = ˆ d , ˆ d ( x ) = ˆ d ( x ) ; b) if f K ( x ) = f I ( x ) , then by (6.37) and f K = ˆ d , ˆ d ( x ) c ≥ (for x ∈ B ). In both cases a) and b), (6.38) holds (by M ≥ an upper bound of ˆ d and c , c > ).Finally, Since the same inequalily of (6.38) holds for all x outside B by definition of B (with ˆ d = ˆ d = 0 for x ∈ X ∪ X ), the equivalence between ˆ d and ˆ d (for all x ∈ X ) is proved.The proof for the general case clearly flows from the similar pattern as above (although tedious).We shall omit the detail. (cid:3) 7. T RACE INTEGRALS AND P ROOF OF T HEOREM A setup, including a comparison with recent developments. There is a vast literature aboutheat kernels on manifolds. A comparison between the previous results and those of ours in the presentpaper shall now be discussed before we proceed further. A concise account of the (ordinary) heatkernel in diversified aspects is given in Richardson [57] and references therein. A generalization ofthe heat kernel to orbit spaces of a group Γ (of isometries) acting on a manifold M dates back to theseminal work of H. Donnelly in late ’70s [21], [22]. Among others, Donnelly calculated the asymptoticexpansion of the trace of the ordinary heat kernel on M restricted to Γ -invariant functions (here Γ -action is assumed to be properly discontinous on M ). Br¨uning and Heintze in ’84 [11] studied theequivariant trace with Γ replaced by a compact group G of isometries (including the trace restricted to G -invariant eigenfunctions). A similar study (of trace) into the orbifold case has been made recentlyin [57] and [20]. In all of these works the asymptotic expansion of the (ordinary) heat kernel is moreor less regarded as known. The questions or techniques come down partly to that used in Donnelly[21] where the contributions to the trace integral are shown to be essentially supported on the fixedpoint set of the group action.In a closely related direction some authors consider the case of Riemannian foliations. In thisregard, if the orbits of a group acting by isometries are of the same dimension, this forms an exampleof a Riemannian foliation. For a Riemannian foliation, one is usually restricted to the space of basic EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 70 functions which are constant on leaves of the foliation. Similar ideas apply to give basic forms . The basic Laplacian and basic heat kernel K B ( t, x, y ) can then be defined. Over decades there has beenmuch study into the existence part of the basic heat kernel K B ( t, x, y ) , which is finally proven in greatgenerality by E. Park and K. Richardson in ’96 [54]. Another proof on the existence is found in ’98[56], which gives a specific formula for K B ( t, x, y ) and allows them to obtain an asymptotic formulafor K B ( t, x, x ) . We denote the trace integral (on basic functions) by Tr e − t △ B (which is P m e − tλ m forcertain eigenvalues with multiplicities). In [57] and [56] the trace integral is also denoted by K B ( t ) which will be avoided here due to a possible confusion. We shall dwell upon this important point afterthe next paragraph.Let’s first pause for a moment for comparison. For the part of the trace integral , the basic techniquebased on Donnelly is also employed here so that the extra contributions, if exist, are expected to besupported on the (lower dimensional) strata. One of our features, however, is Lemma 7.6 which leadsto a precise information about the Gaussianlike term of the heat kernel and facilitates our ensuingasymptotic expansion (of the trace integral) in explicit expressions essentially based only on the datagiven by the ordinary (Kodaira) heat kernel (hence computable in a sense, cf. Remarks 7.25, 1.9). Inthe process we also need to sum over the group elements (Subsections 7.2, 7.3) and patch up theselocal sums over X (Subsections 7.4, 7.5). For the part of the asymptotic expansion , our present heatkernel by its very definition is similar to the K B above. Yet objects beyond the basic forms, allowing ageneralization in the equivariant sense, indexed by m ( ∈ Z ) in our notation (with m = 0 correspondingto the case for K B ), with bundle-values, are considered here. Since we allow CR nonK¨ahler case,suitable Spin c structure in our CR version need be devised and equipped here in order for the rescalingtechnique of Getzler and our discovery of the off-diagonal estimate (Theorem 5.9) to go through. Inthis regard, it is not obvious at all (to us) whether the existence theory in the Riemannian case asabove can be directly applied to our case. Indeed, besides the need of the Spin c structure, our proofof the heat kernel is heavily based on the feature of the group action on CR manifolds, encoded bythe BRT trivialization (Subsection 2.4), through the use of the adjoint version of the original equation(Subsection 1.7). Above all, it lies in the following how our approach distinguishes itself from thoseof others.Notably, a seeming inconsistency could occur. That is, a discovery in the works [57] and [56] revealsthat the so obtained asymptotic expansion for K B ( t, x, x ) there cannot be integrated (over x ) to givethe asymptotics of the trace (integral). This perhaps takes one by surprise. See p. 2304 of [57] andRemark in p. 379 of [56]. Despite this, the work [57] manages to prove an asymptotic expansion forthe trace integral (on basic functions) by using the work [11] (rather than by integrating the asymp-totics of K B ( t, x, x ) obtained therein). In this way, some nontrivial logarithmic terms are to appearunless they are proved to be vanishing. A conjecture has thus been introduced by K. Richardson in ’10[57, Conjecture 2.5] to the effect that in the Riemannian setting as above, for the (special) case of theisometric group action on a compact manifold, the logarithmic terms in the asympototic expansion ofthe trace integral Tr e − t △ B must vanish and under a mild assumption (on orientation), there shall beno fractional powers in t (except possibly an overall fractional power in t ). It is worth mentioning thatthe works [56] and [57] discuss a number of interesting examples pertinent to the aforementionedpeculiar phenomenon. Despite that the seeming inconsistency is consistent with examples by explicitcomputations, it remains conceptually unclear how this phenomenon comes about.Our present work affirms the above conjecture of Richardson (with extension to the S -equivariantcase) in the special case of CR manifolds studied here (see Theorems 7.20, 1.14). One key point forall of this lies in (1.4) with t -dependent coefficients in t powers, which is regarded as the asymptoticexpansion one shall be dealing with in this paper, rather than a classical looking one (1.3) (which issimilar in nature to those proposed and studied in [56], [57]). See also our Remarks 1.6, 1.7 and 1.8,which are closely related to the above singular behavior of a classical formulation of asymptotic ex-pansion. Put simply, the formulation (1.3) of an asymptotic expansion leads to certain discontinuities EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 71 of the t -coefficients along the strata (cf. [56, (4.7)] for a concrete example). A remedy for (1.3) by(1.4) is mainly made via the introduction of a “distance function” (see Theorem 1.3). Eventually, inthis work we can restore the trace integral as the integration of our (unconventional) asymptotic ex-pansion of the relevant heat kernel (see Definition 5.4 for the meaning of our asymptotic expansion).Thus, our trace integral and our asymptotic expansion of the heat kernel jointly clarify (with our classof manifolds) the somewhat undesirable phenomenon which is as mentioned above.To go from the trace integral to the index theorem (thought of as a supertrace integral) is usuallynot immediate. To the knowledge of the authors, the argument for the proof of index theorems byusing trace integrals remains unclarified (cf. Remark 7.26). Completely new ideas might be required;see [12], [13] for very interesting ideas. In the present paper, we couldn’t make our understanding ofthe (transversal) heat kernel (for our class of CR manifolds) complete without employing the rescalingtechnique of Getzler and the off-diagonal estimate (Theorem 5.9) adapted to our setting. These resultsexplore in depth the non-Gaussian terms of our (transversal) heat kernel, in contrast to the Gaussian-like term explored in the trace integral here. With these two parts together, our approach studies themeaningful separate aspects of the heat kernel in an unified manner, hence results in an (local) indextheorem and the trace integral. These point to the differences between our approaches/results andthose of the recent development.We turn now to our proof of the trace integral. The line of thought in the proof involves four stages.In the first stage while the proof in the beginning echos that in last section, we shall make use ofLemma 6.6 and Theorem 6.7 to handle the distance function ˆ d . (Here we assume the strongly pseu-doconvex condition on X .) After this initial step, we shall take a different approach that supersedesthe previous one, which is more quantitative in nature without the strongly pseudoconvex conditionon X (hence without using Lemma 6.6 and Theorem 6.7). This approach is partly based on the dif-ferential geometric information of the various isotropy actions associated with the fixed point sets(strata) of the S action. This allows us to learn more precise details about the heat kernel of KohnLaplacian, hence to refine the computation in (7.8) which is basically qualitative. (See (7.8) for a kindof Dirac delta functions associated with the strata.) Remark that one key point here is the notion of type which is initially designed for the need of computation. In the fourth stage it is attached to the S stratification closely.In the second, third and fourth stages, the treatment goes in line with that in the first stage and ismostly technical so as to integrate the results obtained in the first stage in a well organized manner.The nonuniqueness way (subject to choice of BRT trivializations) of giving the asymptotic expansionof e − t e (cid:3) + b,m ( t, x, y ) (cf. Theorem 6.1) leaves us the freedom of choosing convenient BRT charts to workout some computations. The salient fact that e − t e (cid:3) + b,m ( t, x, y ) is an intrinsic object (yet not directlycomputable), thus is independent of choice of BRT trivializations, is essential to giving intrinsic mean-ings to some BRT-dependent computations (cf. the contrast between Propositions 7.16 and 7.18 on η s -terms). This conceptual understanding turns out to be crucial to our final result. The extension ofthe previous notion “type” to the S stratification is the last conceptual step for the completion of theproof.As before, X (dim X = 2 n + 1) is a compact connected CR manifold with a transversal CR locallyfree S action. To proceed with the proof of Theorem 1.14, assume X = X p S X p S · · · S X p k where X p ℓ = S s ℓ γ =1 X p ℓ ( γ ) ( s ℓ =1 = 1) as a disjoint union of (connected) submanifolds X p ℓ ( γ ) ( X p ℓ , being thefixed point set of an isometry e − i πpℓ , is a submanifold (possibly disconnected)).Write e ℓ ( γ ) for the (real) codimension of X p ℓ ( γ ) in X . When there is no danger of confusion, we maydrop γ and write e ℓ for e ℓ ( γ ) . Recall X ℓ − = X p ℓ S X p ℓ +1 · · · .We follow the notations in Subsection 5.2 and the beginning of the last section. Thus B j :=( D j , ( z, θ ) , ϕ j ) ( j = 1 , , . . . , N ) with D j = U j × ] − δ j , e δ j [ , U j = { z ∈ C n ; | z | < γ j } and similarly EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 72 ˆ D j = ˆ U j × ] − δ j , e δ j [ , ˆ U j = (cid:8) z ∈ C n ; | z | < γ j (cid:9) . We let δ j = e δ j = ζ , j = 1 , , . . . , N and assume X = ˆ D S · · · S ˆ D N . As before, we assume that ζ > satisfies (1.15).7.2. Local angular integral. Recall ˆ h j, + ( x, y ) , ˆ b + j,s ( x, y ) of (5.37) (to be given below); a + s ( t, x, y ) involves a certain integral over [0 , π ] (cf. (5.41)), s = n, n − , . . . . One key step is the following localversion. That is, the (trace) integral of the form(7.1) I = I ( j ) ( p ℓ , g ( x )) ≡ π Z πpℓ + ε πpℓ − ε Z X g ( x ) e − ˆ hj, +( x,e − iu ◦ x ) t Tr ˆ b + j,s ( x, e − iu ◦ x ) e − imu dv X ( x ) du. The trace “Tr” here is actually well defined despite a slight abuse of notation about ˆ b + j,s ( x, e − iu ◦ x ) atthe second variable (see the line above (2.2)).Recall the expressions in (5.37) (to be used in what follows): ˆ h j, + ( x, y ) = ˆ σ j ( θ ) h j, + ( z, w )ˆ σ j ( η ) ∈ C ∞ ( D j ) , x = ( z, θ ) , y = ( w, η )ˆ b + j,s ( x, y ) = χ j ( x ) e − mϕ j ( z ) − imθ b + j,s ( z, w ) e mϕ j ( w )+ imη τ j ( w ) σ j ( η ) , s = n, n − , . . . (7.2)with suitable cut-off functions χ j , τ j , σ j and ˆ σ j defined there.There will be cases for the result (7.1). We need some preparations and notations.For I = I ( p ℓ , g ) of (7.1), take a point x ∈ Supp g ∩ X p ℓ , then x ∈ X p ℓ ( γℓ ) for a γ ℓ = 1 , . . . , s ℓ .Locally at x there are higher dimensional strata X p i γi = X ) X p i γi ) . . . ) X p if ( γif ) ) X p if +1( γif +1 ) = X p ℓ ( γℓ ) passing through x where i = 1 < i < . . . < i f < i f +1 = ℓ, ∈ { , , , . . . , ℓ − , ℓ } . Here (to be usefullater) p i | p i · · · | p i f | p ℓ (by Remark 1.16 similarly). We say Definition 7.1. i) The type τ ( I ) of I ( p ℓ , g ) is τ ( I ) := ( i ( γ i ) , i ( γ i ) , . . . , i f ( γ i f ) , i f +1 ( γ i f +1 )) where i = γ i = 1 and i f +1 = ℓ always. The length l ( τ ( I )) of the type is f + 1 . I ( p ℓ , g ) is said to be of simpletype if in τ ( I ) , ( i , i , . . . , i f +1 ) = (1 , , . . . , ℓ − , ℓ ) .ii) Two given types τ ( I ( p ℓ , g )) = ( i ( γ i ) , i ( γ i ) , . . . , i f +1 ( γ i f )) τ ( I ( p ℓ , g )) = ( j ( γ ′ j ) , j ( γ ′ j ) , . . . , j f +1 ( γ ′ j f )) are said to be in the same class provided a) f = f := f , ℓ = ℓ , i = j , i = j , . . . , i f = j f and b)the codimensions of the corresponding strata coincide: e ℓ ( γ ℓ ) = e ℓ ( γ ′ ℓ ) , e i ( γ i ) = e j ( γ ′ j ) , e i ( γ i ) = e j ( γ ′ j ) , . . . , e i f ( γ if ) = e j f ( γ ′ jf ) .iii) As above I = I ( p ℓ , g ) , suppose Supp g ∩ X p ℓ = ∅ , equivalently Supp g ⊂ ∪ ℓ − q =1 X p ℓ − q . We say τ ( I ) is of trivial type .Remark that g ( x ) will be chosen to be of very small support and the local nature of I , τ ( I ) willbe obvious. Namely, in this case τ ( I ) is independent of choice of x ( ∈ Supp g ∩ X p ℓ ) . In the finalsubsection, the notion of “type” will be naturally extended to each connected submanifold X p ℓ ( γ ) inthe strata. By this, the influence of the geometry of the S stratification on the heat kernel traceintegral will become more evident.Most numerical results in what follows will only depend on the equivalence classes of types. But forthe sake of notational convenience, we assume I to be of simple type or trivial type in the propositionbelow. The modification to the general type is basically only complicated in notation and will betreated later. Proposition 7.2. Suppose x ∈ ˆ D j . Then there exist a neighborhood ˜Ω ( ⋐ ˆ D j ) of x and an ˜ ε > (depending on x ) such that for every Ω ⊂ ˜Ω , every g ( x ) ∈ C ∞ (Ω) we have the following for I of (7.1) with any ε ≤ ˜ ε . Note I is assumed to be of simple type (if not of trivial type) as said prior to the EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 73 proposition. (In the following, ii) and Case a) of iii) are basically of trivial type; i) and Case b) of iii) areof simple type.)i) ℓ = 1 ( p = p ). For x = ( z, v ) ∈ ˆ D j write z ( x ) = z and θ ( x ) = v . I = e − πip m π Z ε − ε Z X g ( x ) χ j ( x )Tr b + j,s ( z, z ) τ j ( z ) σ j ( v + ψ ) dv X ( x ) dψ. In particular, I is a constant independent of t . (Note it is b + j,s instead of ˆ b + j,s here; the same can be saidwith (7.3) below.)ii) Suppose e − i πpℓ ◦ x ˆ D j (here ℓ = 2 , , . . . , k ). Then I = 0 .iii) Suppose e − i πpℓ ◦ x ∈ ˆ D j (here ℓ = 2 , , . . . , k ).Case a) x ∈ S q = ℓ − q =1 X p ℓ − q . Then I ∼ O ( t ∞ ) as t → + .Case b) x S q = ℓ − q =1 X p ℓ − q and x ∈ X p ℓ ( γℓ ) ⊂ X p ℓ . Take local coordinates ( e ℓ = e ℓ ( γ ℓ ) for some γ ℓ = 1 , . . . , s ℓ ) y = ( y , . . . , y n +1 ) = (ˆ y, Y ) with ˆ y = ( y . . . , y e ℓ ) and Y = ( y e ℓ +1 , . . . , y n +1 ) defined on Ω such that X p ℓ ∩ Ω = { y ∈ Ω; y = · · · = y e l = 0 } . Assume (possibly after shrinking Ω about x ) Ω = S j ∈{ , ··· ,k } ( X p j ( γj ) ∩ Ω) (for some γ j = 1 , · · · , s j )which is seen to be (by assumption of simple type) ( X p ℓ ( γℓ ) ∩ Ω) ℓ − [ q =1 ( X p ℓ − q ( γℓ − q ) ∩ Ω) . Write e ℓ − q +1 − e ℓ − q for the codimension of X p ℓ − q +1 in X p ℓ − q where p µ = p µ ( γ µ ) for µ = ℓ − q + 1 and µ = ℓ − q respectively. If y = ( z, θ ) (in BRT coordinates), write z ( y ) for z and if y = (0 , Y ) , write Y for (0 , Y ) and z ( Y ) for z ( y ) . Similar notation for θ ( Y ) etc.Then ( e ℓ = e ℓ ( γ ℓ ) ) I = b ( j ) s, eℓ t eℓ + b ( j ) s, eℓ +12 t eℓ +12 + . . . where the first coefficient b ( j ) s, eℓ is given by b ( j ) s, eℓ = π eℓ e − πipℓ m ℓ − Y q =1 (cid:12)(cid:12)(cid:12) e i πpℓ p ℓ − q − (cid:12)(cid:12)(cid:12) − ( e ℓ − q +1 − e ℓ − q ) × π Z ε − ε Z X pℓ ( γℓ ) g ( Y ) χ j ( Y )Tr b + j,s ( z ( Y ) , z ( Y )) τ j ( z ( Y )) σ j ( θ ( Y ) + u ) dv X pℓ ( γℓ ) ( Y ) du. (7.3) In particular, for s = n (cf . dim X = 2 n + 1) , (7.3) for b ( j ) n, eℓ simplifies by using Tr b + j,n ( z, z ) ≡ (2 π ) − n .Proof. Write x = ( z , θ ) . For simplicity, assume θ = 0 without loss of generality (cf. the last threeparagraphs of the proof of Theorem 6.1 for a similar situation). Note that the existence of ˜Ω and ˜ ε inthe statement above will be obvious from the proof below and we shall not refer to them explicitly.To see i), we note that e − i πp = id ( p = p ) on X (because it is so on X p by definition which is dense(and open) in X ). For x = ( z, v ) lying in the BRT neighborhood ˆ D j and for u = πp ± ε such that e − iu ◦ x = e ± iε ◦ x lies in D j ( ⊃ ˆ D j ), one has e ± iε ◦ x = ( z, v ∓ ε ) by construction of BRT charts D j . Inthis case e − ˆ hj, +( x,e − iu ◦ x ) t ≡ since ˆ h j, + ( x, e − iu ◦ x ) = 0 by h j, + ( z, z ) = 0 of (7.2). The same reasoningapplies to Tr ˆ b + j,s ( x, e − iu ◦ x ) to reach Tr b + j,s ( z, z ) . Now choose a neighborhood Ω ⋐ ˆ D j of x then asmall ε > (depending on x ) such that e ± iε ◦ x lies in D j for x ∈ Ω . As g ∈ C ∞ (Ω) , we can apply theabove argument for these x by making ψ = u − πp ( | ψ | ≤ ε ) so that e − iu ◦ x = e − iψ ◦ x = ( z, v + ψ ) . In EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 74 (7.2) one thus has θ = v , w = z and η = v + ψ . By these remarks, i) of the proposition follows from(7.1) and (7.2).For ii) of the propostion, with x ∈ ˆ D j yet e − i πpℓ ◦ x ˆ D j , by continuity of S action there exist aneighborhood Ω ( ⋐ ˆ D j ) of x and an ε > such that e − iu ◦ x ˆ D j for x ∈ Ω and u ∈ ] πp ℓ − ε, πp ℓ + ε [ .Hence for these x ∈ Ω , ˆ b j, + ( x, e − i πpℓ ◦ x ) = 0 by a cut-off function τ j (of compact support in ˆ U j ⊂ ˆ D j )involved in ˆ b j, + (see (7.2)), giving I = 0 in (7.1).For case a) of iii), the assumption gives x ∈ X p ℓ − q , q ∈ { , , . . . , ℓ − } . Further, by assumption e − i πpℓ ◦ x ∈ ˆ D j we write e − i πpℓ ◦ x = ( e z , e θ ) with (cid:12)(cid:12)(cid:12)e θ (cid:12)(cid:12)(cid:12) < ζ . We claim ˜ z = z . The line of argumentis slightly different from that in (6.4). Suppose ˜ z = z . Then by e i e θ ◦ ( e − i πpℓ ◦ x ) = e i e θ ◦ ( e z , e θ ) =( e z , 0) = ( z , 0) = x (recall θ = 0 in the beginning of the proof). Hence,(7.4) πp ℓ − e θ = m πp ℓ − q , m ∈ Z by assumption x ∈ X p ℓ − q . But (cid:12)(cid:12)(cid:12)e θ (cid:12)(cid:12)(cid:12) < ζ and ζ is assumed to satisfy (1.15), so the above equality isabsurd, proving the claim ˜ z = z by contradiction.Now that e z = z , there exists a neighborhood Ω of x and an ε > (dependent on x ) such that for x ∈ Ω and θ ∈ ] πp ℓ − ε, πp ℓ + ε [ , writing e − iθ ◦ x = ( e z, ˜ θ ) and x = ( z, θ ) one has | e z − z | ≥ | e z − z | ≡ δ by using continuity of S action at x = x and θ = πp ℓ . From the property of ˆ h j, + ( x, y ) (which isessentially | z − w | , cf. (5.19) and (5.37)) one sees that I of (7.1) gives(7.5) π Z πpℓ + ε πpℓ − ε Z X g ( x ) e − ˆ hj, +( x,e − iu ◦ x ) t Tr ˆ b + j,s ( x, e − iu ◦ x ) e − imu dv X ( x ) du = O ( t ∞ ) , as t → + (for g ∈ C ∞ (Ω) ) simply because the exponential term in (7.5) decays rapidly if | e z − z | ≥ δ here,proving case a) of iii) of the proposition.To prove case b) of iii), we first give an estimate under an additional assumption that X is stronglypseudoconvex, then we will drop this assumption and carry out some refined computations to com-plete the proof.Since x is a fixed point of e − πipℓ by assumption, by continuity of S action there exist an opensubset Ω of x and a small constant < ε < ζ such that e − iθ ◦ x ∈ ˆ D j for x ∈ Ω and θ ∈ ] πp ℓ − ε, πp ℓ + ε [ .We assume that Ω is small, say contained in the BRT chart ˆ D j , and satisfies the local coordinates ofcase b) of iii). For x = ( z, v ) ∈ Ω and θ ∈ ] πp ℓ − ε, πp ℓ + ε [ , write e − iθ ◦ x = ( e z, e v ) ∈ ˆ D j .We claim that there exists a positive continuous function f ( x ) such that(7.6) h j, + ( z, e z ) ≥ f ( x ) d ( x, X p ℓ ) , ∀ x ∈ Ω where h j, + is as in (7.2) (cf. (5.19)). (Here is the only place where we use the assumption X isstrongly pseudoconvex.)Granting the claim (7.6), with the local coordinates in iii), suppose y ( x ) = Y ( x ) = 0 . Rewrite thequotient d ( y,X pℓ ) ( | y | + ··· + | y eℓ | ) as(7.7) d ( y, X p ℓ ) = f ( y )( | y | + · · · + | y e ℓ | ) , ∀ y ∈ ˆΩ where f ( y ) is a positive continuous function. EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 75 With (7.6) and (7.7), we estimate ˆ h j, + below and have the following (see (7.2) or (5.37) and note g ( x ) ∈ C ∞ (Ω) , Ω small). I = 12 π Z πpℓ + ε πpℓ − ε Z X g ( x ) e − ˆ hj, +( x,e − iu ◦ x ) t Tr ˆ b + j,s ( x, e − iu ◦ x ) e − imu dv X ( x ) du ≤ π Z πpℓ + ε πpℓ − ε Z X g ( x ) e − f x ) d ( x,Xpℓ )2 t Tr ˆ b + j,s ( x, e − iu ◦ x ) e − imu dv X ( x ) du = 12 π Z πpℓ + ε πpℓ − ε Z X g ( y ) e − f y ) f y )( | y | ··· + | yeℓ | t Tr ˆ b + j,s ( y, e − iu ◦ y ) e − imu dv X ( y ) du ∼ c ( j ) s, eℓ t eℓ + c ( j ) s, eℓ +12 t eℓ +12 + · · · as t → + , (7.8)where the last step is obtained by a change-of-variable (rescaling y i by √ ty i , i = 1 , . . . e ℓ , e ℓ ≥ as ℓ ≥ ) and c ( j ) s.k ∈ R is independent of t ( k = e ℓ , e ℓ +12 , . . . ).We are left with the proof of the claim (7.6). Part of the argument echos that for (6.4). We firstestimate | z − e z | . Without the danger of confusion we omit “ ◦ ” in what follows. By ( z, 0) = e iv x and (˜ z, 0) = e i ˜ v ( e − iθ x ) , | z − e z | is equivalent to d ( e iv x, e i ˜ v ( e − iθ x )) (cf. (6.3)) which is the same as d ( x, e − iv ( e i ˜ v ( e − iθ x ))) . As (˜ z, ˜ v ) , ( z, v ) ∈ ˆ D j , one has ˜ v, v ≤ ζ . By choosing ζ , ε to be (much) less thanthe ε of Lemma 6.6, one sees d ( x, e − iv ( e i ˜ v ( e − iθ x ))) ≥ ˆ d ( x, X ℓ − ) of Lemma 6.6. By the same lemma(7.9) ˆ d ( x, X ℓ − ) is equivalent to ˆ d ( x, X ℓ − ) . As | z − e z | is also equivalent to h j, + ( z, ˜ z ) of (7.6) (cf. (6.3)) and (7.9) is equivalent to d ( x, X ℓ − ) byTheorem 6.7 with our assumption X is strongly pseudoconvex, we have now shown(7.10) h j, + ( z, ˜ z ) ≥ c · d ( x, X ℓ − ) for some constant c > . In view that X ℓ − = X p ℓ ∪ X p ℓ +1 · · · and the assmption that x ∈ X p ℓ , onesees, possibly after shrinking Ω , d ( x, X ℓ − ) = d ( x, X p ℓ ) = d ( x, X p ℓ ) . Hence we have reached (7.6)from (7.10), as desired.Following (7.8) we shall now make some accurate computations for case b) of iii) of this propostion.Henceforward we do not assume that X is strongly pseudoconvex; we will not use Lemma 6.6 andTheorem 6.7 as used above.Write πp ℓ = ω and u = ψ + ω . In coordinates of case b) of iii), in view of (7.8) one seeks to identify,among others,(7.11) lim t → + ˆ h j, + (( √ t ˆ y, Y ) , e − iu ( √ t ˆ y, Y )) t where we have rescaled ˆ y → √ t ˆ y , and we omit “ ◦ ” for the e − iu ∈ S action.Since the fixed point set of an isometry is totally geodesic, we assume Y to be a system of geodesiccoordinates at Y = 0 of X p ℓ , as (ˆ y, Y ) the geodesic coordinates at (0 , of X . We choose Y = 0 in(7.11) to simplify the notation. Expressed in BRT coordinates, ( √ t ˆ y, 0) = ( z , v ) and e − iω ( √ t ˆ y, 0) =( z , v ) (by continuity of S action, for t small e − iω ( √ t ˆ y, ∈ ˆ D j since e − iω (0 , 0) = (0 , ).One sees e − iu ( √ t ˆ y, 0) = e − iψ ( z , v ) = ( z , v + ψ ) for | ψ | ≤ ε . By (7.2) one sees(7.12) ˆ h j, + (( √ t ˆ y, , e − iu ( √ t ˆ y, is now reduced to h j, + ( z , z ) which is independent of ψ for u in the ε -neighborhood of ω . Namely ˆ h j, + (( √ t ˆ y, , e − iu ( √ t ˆ y, ≡ ˆ h j, + (( √ t ˆ y, , e − iω ( √ t ˆ y, for u in the ε -neighborhood of ω .Now x = (0 , is a fixed point of e − iω ( ω = πp ℓ ). One can see that T x X under the isotropy actioninduced by e − iω decomposes as an orthogonal direct sum of eigenspaces (where N ( S/M ) denotes the EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 76 normal bundle of a submanifold S in an ambient manifold M with N p ( S/M ) the fiber of N ( S/M ) at p , and X p µ = X p µ ( γµ ) for some γ µ = 1 , , · · · , s µ ; µ = 1 , , · · · , ℓ ) T x X p ℓ , N x ( X p ℓ /X p ℓ − ) , N x ( X p ℓ − /X p ℓ − ) . . . , N x ( X p /X p ) associated with eigenvalues(7.13) , e iωp ℓ − , e iωp ℓ − , . . . , e iωp respectively.For instance, assume ℓ = 2 and take g = e πip . Set q = p p ( ∈ N ) . On N x ( X p /X p ) , g = id and g q = id .Hence v ∈ N x ( X p /X p ) is rotated by the angle πq which is ωp .The goal in what follows is to prove the claim that for q = 1 , . . . , ℓ − ,(7.14) lim t → + ˆ h j, + (( √ t ˆ y, , e − iω ( √ t ˆ y, t = (cid:12)(cid:12) e iωp ℓ − q − (cid:12)(cid:12) || ˆ y || for ˆ y ∈ N x ( X ℓ − q +1 /X ℓ − q ) or equivalently, in the notation above (see (7.12))(7.15) lim t → + h j, + ( z , z ) t = (cid:12)(cid:12) e iωp ℓ − q − (cid:12)(cid:12) || ˆ y || (for ˆ y ∈ N x ( X ℓ − q +1 /X ℓ − q ) )where || · || denotes the norm with respect to the metric tensor of X at x .Our proof of claim (7.14) is based on the following sequence of lemmas. Lemma 7.3. In the notation above, for ˆ y ∈ N x ( X ℓ − q +1 /X ℓ − q ) we have lim t → + d X (( √ t ˆ y, , e − iω ( √ t ˆ y, t = (cid:12)(cid:12) e iωp ℓ − q − (cid:12)(cid:12) || ˆ y || where d X denotes the distance on X .Proof. On T (0 , X the action induced by e − iω rotates the tangent vector ˆ y by the angle ωp ℓ − q . Hencethe lemma follows from the well known fact that in a Riemannian manifold ( M, g ) , if a , b in M arethe images of A , B in T p M by the exponential map at p ∈ M , then as ( a, b ) → ( p, p ) (7.16) lim d M ( a, b ) || A − B || → where || · || is g at p (cf. [39, Proposition 9.10]). (cid:3) Sublemma 7.4. Suppose N is a Riemannian submanifold of a Riemannian manifold M . Then therespective distance functions on M and on N are infinitesimally the same. More precisely, suppose in N , p n = q n for all n ∈ N , and p n , q n → p as n → ∞ for a given point p ∈ N . Then lim n →∞ d M ( p n ,q n ) d N ( p n ,q n ) = 1 .Moreover, suppose t n,M (resp. t n,N ) in T p n M are the unit tangent vectors along which the minimalgeodesics in M (resp. N ) join p n and q n . Then lim n ( t n,M − t n,N ) = 0 .Proof. Suppose the special case p n = p for all n . Let γ n be a geodesic (with unit speed) of N joining p and q n , and β n = exp − p ( γ n ) ⊂ T p M . Write l n ( t ) for the length of (part of) β n (with the parametergoing from to t ) measured with the metric g ij = 1 + O ( | x | ) in geodesic coordinates (at p ). Write || v || for the Euclidean norm of a vector v ∈ T p M expressed in geodesic coordinates. Given a curve β ( t ) ⊂ T p M , β (0) = p , ˙ β (0) = 0 , one sees the length function l ( t ) = R t q < ˙ β ( t ) , ˙ β ( t ) > g ij dt satisfies (cid:12)(cid:12)(cid:12) || β ( t ) ||− l ( t ) || β ( t ) || (cid:12)(cid:12)(cid:12) = O ( t ) ≤ Ct for a (locally bounded) quantity C which depends only, apart from β , onthe local geometry at p (uniformly). Clearly this implies the lemma if q n is assumed to approach p along a given geodesic γ of N . If q n approaches p along different geodesics γ n , since these geodesicscan be uniformly controlled by the local geometry around p , the same results hold as well. For thegeneral case where p n are different, the similar argument using the control by local geometry implies (cid:12)(cid:12)(cid:12) d M ( p n ,q n ) − d N ( p n ,q n ) d M ( p n ,q n ) (cid:12)(cid:12)(cid:12) ≤ C ( d M ( p n , q n )) . The assertion about the unit tangent vectors can be provedsimilarly. (cid:3) EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 77 Sublemma 7.5. Let N be a differentiable manifold equipped with two Riemannian metrics g and h , and p ∈ N . Assume that g ( p ) = h ( p ) . Suppose that in N , p n = q n for all n ∈ N such that lim n p n = lim n q n = p . Then lim n →∞ d g ( p n ,q n ) d h ( p n ,q n ) = 1 where d • denotes the (metric-dependent) distance function on N .Proof. The result is local; assume N ⊂ R n as an open subset. By comparison to a fixed Euclideanmetric, we assume g is Euclidean inherited from R n . By reasoning similar to the previous sublemma,it is seen that for n >> , d h ( p n , q n ) is basically (1 ± C max {|| p n − p || , || q n − p ||} ) d g ( p n , q n ) (where || · || denotes the Euclidean norm) with a uniform bound C . The assertion follows. (cid:3) The last lemma (as our main lemma) is as follows. (This lemma can be viewed as a sharp version ofthe important claim (6.4) in the proof of Theorem 6.1, which bears upon the reason why our distancefunction ˆ d arises.) Lemma 7.6. In the previous notation, write p = ( z ( p ) , θ ( p )) and q = ( z ( q ) , θ ( q )) in ( z, θ ) coordinates onthe BRT chart ˆ D j = ˆ U j × ] − ζ , ζ [ . We omit the subscript j in what follows. Let S = S ◦ p be the S -orbitof p and N ( S/X ) be the normal bundle of S in X identified with the orthogonal complement of T S in T X | S . Suppose p n = q n for all n and p n , q n → p ∈ X as n → ∞ such that D n = exp − X,p ( p n ) , A n =exp − X,p ( q n ) ∈ N p ( S/X ) . In the case where p n = p and q n = p (all n ), suppose the angle at p given by thevectors D n and A n are bounded away from as n → ∞ . Then (see (4.1) for the metric on U giving d U below) (7.17) lim n →∞ d X ( p n , q n ) d U ( z ( p n ) , z ( q n )) = 1 . In particular z ( p n ) = z ( q n ) for n large.Proof. In this proof we take the same notation U = ˆ U j seated as an embedded submanifold of X with θ = 0 . As in (the proof of) Sublemma 7.4, we first assume p n = p for all n . By applying the S isometries we assume p = ( z ( p ) , ∈ U . By using the construction of our rigid metric on X (cf. (2.5)and (4.1)) it may be assumed that at p , N p ( S/X ) = T p U . To see this, by the geometrical interpretationof BRT transformations in (the proof of) Proposition 4.2 one can adjust the BRT coordinates such that dφ ( p ) = 0 (similar to the well known fact that for a hermitian metric h of a holomorphic line bundle L on a complex manifold, at any given point p one has dh ( p ) = 0 up to a change of local frames of L ).This gives T , p D = { ∂∂z j + i ∂φ∂z j ( z ) ∂∂θ ; j = 1 , , . . . , n } = { ∂∂z j ; j = 1 , , . . . , n } (and T , D = T , D ) (cf. loc. cit. ). It easily follows N p ( S/X ) = T p U as claimed. A word of caution is in order. The (intrinsic)geometrical interpretation for BRT charts (cf. loc. cit. and remarks after (4.1)) shows that the asserted(7.17) is independent of choice of BRT coordinates (on that particular BRT neighborhood). On theother hand, U considered as an embedded submanifold of X as done above does depend on the choiceof BRT coordinates.The first reduction step is as follows. U is endowed with another (Riemannian) metric inherited, asa submanifold, from that of X . This is different from the metric originally defined (on U , cf. (4.1)),yet the two metrics coincide at p as can be seen above. By Sublemma 7.5 it is enough to prove (7.17)with this inherited metric. Without the danger of confusion we shall adopt the same notation d U ( · , · ) for the new distance function in what follows.Fix an n and set q = q n , q ′ = e iθ ( q ) ◦ q = ( z ( q ) , ∈ U . Put A = exp − X,p ( q ) , B = exp − X,p ( q ′ ) ∈ T p X , so || A || = d X ( p, q ) , || B || = d X ( p, q ′ ) (where T p X is equipped with the Euclidean metric || · || in geodesiccoordinates). We are going to prove, as n → ∞ ,(7.18) lim || A |||| B || = 1 . One has d X ( p, q ′ ) /d U ( p, q ′ ) → by Sublemma 7.4. Hence, to prove (7.17) for this special case p n = p is the same as to prove d X ( p, q ) /d X ( p, q ′ ) → which is (7.18) above. EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 78 To see (7.18) (hence (7.17)), we first argue (7.19) below. Let L ⊂ T p X be the line determined by A, B , i.e. L = { A + t ( B − A ); t ∈ R } . Then(7.19) L is approximately orthogonal to A and to B (as n large) . Note that A ∈ N p ( S/X ) = T p U from the condition of the lemma, and that B is nearly lyingon T p U (with a small angle between B and T p U ) by using Sublemma 7.4 on tangents. Let Γ = { e iθ ◦ q } θ ∈ [0 ,θ ( q )] ⊂ X ( θ ( q ) ≥ , say) joining q and q ′ and Γ = exp − X,p Γ ⊂ T p X joining A and B . Recallthat the vector field T induced by the S action is orthogonal to U at p as mentioned above, hence T q ′ Γ ⊥ T q ′ U approximately (as n >> ). On the other hand, by T p S ⊥ T p U and Γ ≈ S (as n ≫ ),one sees by A ∈ T p U that T A Γ ⊥ T p U approximately (as vector subspaces in T p X ). In sum, if q, q ′ are close to p (so T q ′ U close to T p U ), then T A Γ ⊥ T p U , T B Γ ⊥ T p U approximately (cf. the foliationargument below); for this we write Γ ⊥ A, B approximately. We are ready to prove (7.19). Pullingback the S foliation locally around p via the exp X,p in the same way as Γ obtained by Γ , there is afoliation F around p in T p X in which (part of) Γ lies as a leaf. Write p ∈ Γ ( ⊂ exp − X,p S ) ∈ F . As n ≫ , the line L determined by A, B ∈ Γ tends to the tangent line (= T p S ) to Γ at p (since theleaf Γ of F tends to the leaf Γ ). Hence by using the uniform continuity for F around p , L is close tolines ˜ L tangent to leaves ˜Γ of F if ˜Γ are nearby Γ and ˜ L nearby T p S . In particular, L is close to thetangent lines T A Γ , T B Γ (as n ≫ ). Now that Γ ⊥ A, B approximately as just shown, giving readily T A Γ ⊥ A, T B Γ ⊥ B approximately, this in turn yields L ⊥ A, B approximately ( n ≫ , proving(7.19).For q ′ close to p , by simple Euclidean geometry (on T p X ), || A − B || is rather small in comparison to || A || and || B || by using L ⊥ A, B approximately (7.19), i.e. || A − B || = o ( || A || ) , o ( || B || ) . By using lawof cosines, one can obtain (7.18), yielding the special case p n = p of the lemma. As this step appearscrucial and will be instrumental to the general case, we prefer to supply some details as follows.Take a triangle with vertices T i ( i = 1 , , , angles α i at T i and δ i the length of the side facing T i .Suppose α ≤ α and both ≈ π . Set α = π − α , < α ≪ . Let D sit on the line L determined by T and T such that the line L determined by T and D is perpendicular to L . Assume first that D sitsbetween T and T . Then δ = δ sin θ + δ sin θ where θ , θ (with θ + θ = α ) are angles given by L and the two sides at T . Thus δ δ ≤ α ( δ δ ≤ by α ≤ α ). If D sits outside the triangle, then δ = δ sin α − δ sin θ , θ = α − α , so δ δ ≤ sin α . One obtains δ δ → if both α → and α → . By δ = δ + δ − δ δ cos α , one has (1 − δ δ ) = ( δ δ ) − δ δ (1 − cos α ) ≤ ( δ δ ) ≤ sin α + 2 sin α giving δ δ → if both α , α ≈ π (hence α, α ≈ ). As said, this yields (7.18).We draw some consequences in order for the general case. If α → (by α , α → π/ ), then thetwo sides at α are close to each other, i.e. lim( A/ || A || − B/ || B || ) = 0 (as q → p ). One sees that if C = exp − U,p ( q ′ ) ∈ T p U , then by using (7.18) and Sublemma 7.4 on tangents via B ,(7.20) a ) lim || A || / || C || = 1 , b ) lim( A/ || A || − C/ || C || ) = 0 . We are ready to prove the general case p n = p . Write D = exp − X,p ( p n ) , F = exp − U,p ( p ′ n ) in the sameway as A = exp − X,p ( q n ) , C = exp − U,p ( q ′ n ) above. With D, F in place of A, C in (7.20) one has the sameresults for D, F :(7.21) a ) lim || D || / || F || = 1 , b ) lim( D/ || D || − F/ || F || ) = 0 . In view of (7.16) one has || D − A || /d X ( p n , q n ) → , || F − C || /d U ( p ′ n , q ′ n ) → . Hence to prove (7.17),i.e. d X ( p n , q n ) /d U ( p ′ n , q ′ n ) → , is the same as to show lim || D − A || / || F − C || = 1 . This is intuitivelyclear by (7.20), (7.21) (which alludes to A ≈ C and D ≈ F ) provided that the angle given by the twovectors D and A at p (hence by F and C at p , cf. b) of (7.20) and (7.21)) is not approaching zero. EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 79 This is precisely the condition given in the lemma. For the rigor of this argument one may use law ofcosines without difficulty. Hence the lemma follows. (cid:3) proof of claim (7.15) . By combining Lemma 7.6 and Lemma 7.3 we can finish the proof of the claim(7.15) provided that h j ( z , z ) = d U j ( z , z ) . But this is a standard fact for the heat kernels of Diracand Laplacian type (see [6, Theorem 2.29]); see also the famous result of S. R. S. Varadhan[60] for ageneralization in this regard. (cid:3) proof of Proposition 7.2 resumed .We are now ready to prove case b) of iii) of Proposition 7.2. To work on the integral I of (7.1) weare going to refine the computation contained in (7.8). Indeed, case b) of iii) can be obtained if onenotes the following α ) − ǫ ) (part of them similar to the proof of i) of this proposition): α ) R ∞−∞ e − a x dx = √ π | a | ; β ) using (7.14) (with q = 1 , , . . . , ℓ − ) for e − ˆ h + j,st in (7.1); γ ) change of variable u = ψ + πp ℓ in (7.1); δ ) in (7.1), by rescaling ˆ y → √ t ˆ y , ˆ b + j,s (( √ t ˆ y, Y ) , e − iu ◦ ( √ t ˆ y, Y )) replacing ˆ b + j,s ( x, e − iu ◦ x ) , tends to ˆ b + j,s ((0 , Y ) , e − iu ◦ (0 , Y )) = ˆ b + j,s ((0 , Y ) , e − iψ ◦ (0 , Y )) because (0 , Y ) ∈ X p ℓ . But | ψ | being small ( ≤ ε ) , e − iψ ◦ (0 , Y ) does not change the z ((0 , Y )) ( = z ( Y ) ) coordinate in ˆ D j , giving ˆ b + j,s ((0 , Y ) , e − iψ ◦ (0 , Y )) = b + j,s ( z ( Y ) , z ( Y )) (up to cut-off functions); ǫ ) as t → , σ j ( η ) = σ j ( θ ( e − iu ◦ (0 , Y ))) = σ j ( θ ( e − iψ ◦ (0 , Y ))) = σ j ( θ ( Y ) + ψ ) . With η = θ + ψ and u = ψ + 2 π/p ℓ in (7.1) and (7.2), a cancellation occurs for the three exponentials there; eventually anumerical factor e − i πm/p ℓ is pulled out. And instead of Tr ˆ b + j,s in I of (7.1), we are reduced to χ j Tr b + j,s (no “hat” on b + j,s here) as put down in this proposition.The formula for the coefficient b ( j ) s, eℓ of (7.3) follows from α ) − ǫ ) above.Finally, for s = n , it is well-known that (dropping j here) Tr b + n ( z, z ) in the integral (7.3) being theleading coefficient term in the asymptotic expansion of the ( Spin c ) Kodaira heat kernel, is constant in z and equals (4 π ) − n · (cid:0) rk( V T ∗ , ( U )) (cid:1) = (2 π ) − n ([37, (a) of Theorem 4.4.1], cf. [6, Theorem 2.41]). (cid:3) Global angular integral. To work out the global version (i.e. the integration on [0 , π ] ) it isnatural to consider not only (an ε -neighborhood of) π/p ℓ but also all their multiples s π/p ℓ , s ∈ N , s ≤ p ℓ . The analysis will thus partly depend on whether s π/p ℓ = s ′ π/p ℓ ′ for some s, s ′ , p ℓ , p ℓ ′ ornot. One needs a systematic control of the behavior in this regard. Further, since the result will appearas a sum over these ε -neighborhoods, to organize this sum in a manageable way is also desirable. Weshall now mainly deal with these issues in this subsection.There are minor duplication and perhaps discrepancy in notation between this subsection and thepreceding one. But, it would have appeared cumbersome if we had set up this generality right in thepreceding subsection (as the resulting proof would have become much less illuminating).Recall the S -period of X denoted by πp > πp > . . . > πp k with p | p ℓ , ≤ ℓ ≤ k (cf. Remark 1.16)and the stratum X p ℓ (the set of points of period πp ℓ ) is a disjoint union of connected submanifolds S ≤ γ ℓ ≤ s ℓ X p ℓ ( γℓ ) . Definition 7.7. Fix a smooth function g ( x ) on X . We say that g is of small support if the followingconditions are satisfied. i ) Supp g ⊂ X p i γi ∪ X p i γi ∪ . . . ∪ X p it ( γit ) ∪ X p it +1( γit +1 ) , i < i < · · · < i t +1 ≤ kii ) Supp g ∩ X p is ( γis ) = ∅ , ∀ s, ≤ s ≤ t + 1 iii ) X p i γi ) X p i γi ) . . . ) X p it ( γit ) ) X p it +1( γit +1 ) . (7.22) EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 80 Obviously, given any x ∈ X there exists a neighborhood Ω ∋ x such that every nontrivial g ( x ) ∈ C ∞ (Ω) is of small support in the sense above. Definition 7.8. Let g ( x ) be a smooth function on X of small support in the sense above, (7.22). Let c ∈ N . We define a number i ( c, g ) = i ( c ) associated with c and g as follows.i) i ( c, g ) := ℓ ≥ if the following is satisfied a) c | p ℓ , ℓ = i s for some s , ≤ s ≤ t + 1 and b) c p i s ′ for all s ′ < s . ii) i ( c, g ) := 1 if c | p ( p = p i ) . This is independent of g . iii) i ( c, g ) := ∞ if c p i s foreach s with ≤ s ≤ t + 1 .It is easily seen that p i ( c ) | p i s for each i s with i ( c ) ≤ i s ≤ i t +1 if i ( c ) = ∞ . Indeed, p i | p i | . . . | p i t +1 (cf. Remark 1.16 for a similar case).By the above definition, one sees Lemma 7.9. Let x ∈ Supp g , i ( c ) = ∞ and h ∈ N with ( h, c ) = 1 . It holds e − i πhc ◦ x = x if and only if x ∈ X p i ( c ) . Let h ∈ N with ( h, c ) = 1 and h < c . We consider the integral similar to (7.1) for i ( c ) = ∞ :(7.23) I = I ( j ) ( p i ( c ) , g ( x ) , hc ) ≡ π Z πhc + ε πhc − ε Z X g ( x ) e − ˆ hj, +( x,e − iu ◦ x ) t Tr ˆ b + j,s ( x, e − iu ◦ x ) e − imu dv X ( x ) du. The above extends to the case i ( c ) = ∞ simply by formally setting I ( j ) ( p i ( c )= ∞ , g ( x ) , hc ) to be theintegral to the right of (7.23). Definition 7.10. i) Set i ( c ) of Definition 7.8 to be ℓ . Assume ℓ = ∞ . Define the type τ ( I ( p i ( c ) , g ( x ) , hc )) to be τ ( I ( p ℓ , g ( x ))) where τ ( I ( p ℓ , g )) is given in i) of Definition 7.1. ii) The notions of simple type and class are defined similarly. (Note there is no trivial type for ℓ = ∞ .) iii) If i ( c ) = ∞ , then Supp g ∩ X a = ∅ where X a is the fixed point set of a = e − i π/c ∈ S (whether X a is empty or notdepends on c ). In this case we define it to be the trivial type (cf. iii) of Definition 7.1). Remark . The notion of type concerns only the local stratificaton (of the S action) at x aroundwhich Ω ⊃ Supp g is a small neighborhood. Thus, with a small open subset Ω ⊂ X one may associatethe type τ (Ω) without referring to any kind of integral. In the fourth subsection, we will basicallyadopt this viewpoint for our purpose.First, one examines the case i ( c, g ) = ∞ , which turns out to be inessential. Lemma 7.12. Let c ∈ N . There exists a (finite) covering of BRT trivializations on X with the followingproperty. For any smooth function g of small compact support (cf. Definition 7.7), suppose i ( c, g ) = ∞ and x with g ( x ) = 0 , is given. Then there exist a small open set Ω ∋ x and a small ˜ ε > such that forany χ ∈ C ∞ (Ω) , if one replaces g by χg in the integral I of (7.23) , this I with any < ε ≤ ˜ ε , equals ,or O ( t ∞ ) as t → + .Proof. By the definition of i ( c, g ) = ∞ , if the fixed point set X a of a = e − i π/c is empty, one chooses acovering of BRT trivializations { D j } j (cf. lines above Subsection 7.2) such that ˆ D j ∩ a h ( ˆ D j ) = ∅ for all j and h with ( h, c ) = 1 . In this case the remaining argument by using the continuity of the S action,is almost the same as ii) of Proposition 7.2, yielding I = 0 . If X a is not empty, one chooses any (finite)covering of BRT trivializations (such as the one given prior to Subsection 7.2). Then it may occur theextra case a h ◦ x ∈ ˆ D j for some ˆ D j ∋ x if the choice of g is such that x is very near X a . In thiscase the remaining argument is essentially similar to Case a) of iii) of Proposition 7.2, giving rise to I = O ( t ∞ ) as t → + . (cid:3) To compute I of (7.23), we assume the simple type condition for I (when it is not of trivial type) asgiven in Definition 7.10. Combining Lemma 7.12 we have the following corollary, as a generalizationof Proposition 7.2. EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 81 Corollary 7.13. Notations and the simple type condition as above. Assume that the covering by BRTtrivializations satisfies Lemma 7.12. Let c ∈ N , x ∈ X , Ω ∋ x an open subset and g ∈ C ∞ (Ω) (of smallsupport as above). Then the ε > (in I ) and Ω can be chosen to satisfy the following. a) The same results(for computing I = I ( j ) ( p i ( c ) , g ( x ) , c ) of (7.23) ) hold true as in Proposition 7.2 provided that one adoptsthe replacement of e − i π/p , e − i π/p ℓ by e − i π/c in i), ii) and iii) of the statement, and ℓ (not the one in πp ℓ ) by i ( c ) in Cases a), b) of iii) throughout (so p ℓ − q → p i ( c ) − q , e ℓ → e i ( c ) , e ℓ − q +1 → e i ( c ) − q +1 , e ℓ − q → e i ( c ) − q and γ ℓ → γ i ( c ) , X ℓ → X i ( c ) , X ℓ − q → X i ( c ) − q etc. in Case b)). Note that after the replacement, i ( c ) = ∞ in Case a) and i ( c ) = ∞ in Case b), of iii).b) More generally, for h ∈ N , ( h, c ) = 1 and h < c , with the replacement πc → πhc (and ℓ not the onein πp ℓ by i ( c ) in Cases a), b) of iii)), the same results (for computing I = I ( j ) ( p i ( c ) , g ( x ) , hc ) of (7.23) )hold true as well.proof of Corollary 7.13. One sees that with the replacement of πp ℓ by πc or πhc , the condition on c (inDefinition 7.8) renders the argument in proof of Proposition 7.2 essentially unchanged. For instance,with (7.4) replaced by πhc − e θ = m πp ℓ − q , taking ζ smaller does the job. Further, with substitution of ℓ (not in πp ℓ ) by i ( c ) , the distinct eigenvalues { , e i πpℓ p ℓ − , e i πpℓ p ℓ − , . . . e i πpℓ p } of the isotropy action (of e − i πpℓ at x ) (cf. (7.13)) are changed to { , e i πhc p i ( c ) − , e i πhc p i ( c ) − , . . . e i πhc p } (of e − i πhc at x ) (whichremain distinct). (cid:3) Let c ∈ N and g a smooth function on X of small support as above, with i ( c ) (= i ( c, g )) in Defini-tion 7.8. We are going to associate certain numerical factors. For a contrast, we will give them forcases of the simple type and the general type separately (cf. Definitions 7.1 and 7.10).For the simple type, the numerical factor d c = d c,g,m is set to be i ) i ( c ) = 1 if c > , d c (= d c,g,m ) := X h ∈ N ,h Numerical factors d c for the general type. Given τ ( I ) = ( i ( γ i ) , i ( γ i ) , . . . , i f ( γ i f ) , i f +1 ( γ i f +1 )) of general type, I = I ( p ℓ , g ( x ) , hc ) with i ( c ) = ℓ , the numerical factors d c similar to (7.24) are definedas follows.If c is of i ( c ) = 1 or ∞ , then d c (= d c,g,m,I ) is the same as d c in i), iii) of (7.24). For c with ∞ > i ( c ) ≥ , ∞ > i ( c ) (= i ( c, g )) ≥ , d c (= d c,g,m,I = d c,g,m,τ ( I ) = d c,g,m,τ (Ω) ):= ( √ π ) e i ( c )( γi ( c )) X h ∈ N ,h Notations as above with the general type τ ( I ) allowed (cf. Definitions 7.1 and 7.10).Assume that the covering by BRT trivializations { ˆ D j } j satisfies Lemma 7.12. Let c ∈ N and x ∈ X . Foran ε , write λ c := S h ∈ N ,h Suppose we are given any δ > and an s = n, n − , . . . . Then there exists a (finite)covering { ˆ D j } j of X by BRT charts that satisfy Lemma 7.12 and the following. Suppose x ∈ X and Ω aneighborhood of x with Ω ⋐ ˆ D j for every ˆ D j ∋ x . Then there exist an ε > and a choice of Ω abovesuch that for g ( x ) ∈ C ∞ (Ω) (of small support in the sense of Definition 7.7), writing J ( j ) s = J ( j ) s,m ( g ( x )) (for any ˆ D j ∋ x , s = n, n − , . . . as above) one has the following.i) R Λ J ( j ) s du = (cid:0) P p q =1 e − i πqmp (cid:1) I + , ( j ) X,s .ii) Case a) If x ∈ X p , R Λ J ( j ) s du ∼ or O ( t ∞ ) (as t → + ).Case b) If x ∈ X p ℓ with ℓ ≥ , Z Λ J ( j ) s du ∼ (cid:0) X c ≤ i ( c ) ≤ ℓ ( d c I + , ( j ) i ( c )( γ i ( c ) ) ,s √ t e i ( c )( γi ( c )) + O ( √ t e i ( c )( γi ( c )) +1 )) (cid:1) + O ( t ∞ ) (as t → + )where d c = d c,g,m,I , I = I + , ( j ) i ( c )( γ i ( c ) ) ,s , cf. (7.28) and (7.30) .iii) For the part R N J ( j ) s ( g ) du ≡ η ( j ) s ( g ) , the estimate holds | η ( j ) s ( g ) | < δ · R X | g ( x ) | dv X ( x ) .Proof. To see i), by i) and iii) of Corollary 7.15 it suffices to note P c,i ( c )=1 d c = P p q =1 e − i πqmp by(7.24). Case a) of ii) follows directly from i) of loc. cit. whereas Case b) from iii) of loc. cit. by writing R Λ J s du = P ≤ i ( c ) ≤ ℓ R λ c J s du + P i ( c ) ≥ ℓ +1 R λ c J s du .To see iii), note the following. First, the action by e − iθ with θ ∈ N is fixed point free on X . Eachpoint in X has a distinguished neighborhood ˆΩ such that if x ∈ ˆΩ and θ ∈ N , e − iθ ◦ x ˆΩ by usingcontinuity of the S action. By compactness, we can now assume (reset) a (finite) covering of BRTcharts consisting of these ˆΩ and satisfying Lemma 7.12 (as there are only finitely many c here). Given x ∈ X , we take a small neighborhood Ω ⋐ ˆ D j for every ˆ D j ∋ x . One sees that by the cut-offfunction τ j involved in the integrand of J ( j ) s , J ( j ) s ≡ identically (since τ j ( e − iθ ◦ x ) = 0 if e − iθ ◦ x ˆ D j cf. (5.37), which holds for x ∈ ˆΩ = ˆ D j and θ ∈ N ).However, a word of warning is in order. Given the preceding covering of certain BRT charts, ourintegral J ( j ) s shall be computed with respect to this new covering (because as said above, the cov-ering has been “reset”) which we just did. However, it is not necessarily true that the original ε (ofCorollary 7.15 used above) remains altogether applicable in the new setting. Namely, under the newBRT covering we need to choose an ε , possibly smaller than ε , to ensure that the original argument EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 84 of the proof (of Proposition 7.2) go through well. To see what the above means, let’s make the de-pendence on the parameter ε explicit and denote by Λ( ε ) , N ( ε ) etc. The N in the last paragraphwill be denoted by N ( ε ) . Only with the replacement by Λ ( ε ) , Λ ( ε ) , Λ( ε ) ( ⊂ Λ ( ε ) , Λ ( ε ) , Λ( ε ) respectively), N ( ε ) ( ⊃ N ( ε )) and the new BRT covering throughout the present proposition, can weobtain the corresponding ε -versions of i) and ii) of this proposition. And iii) will correspondinglybe replaced by R N ( ε ) J ( j ) s du . But as just shown in the last paragraph, R N ( ε ) J ( j ) s du = 0 , it follows R N ( ε ) J ( j ) s du = R N ( ε ) \ N ( ε ) J ( j ) s du . Now the measure of N ( ε ) \ N ( ε ) is controlled by C · ( ε − ε ) fora fixed constant C ; the integrand of J ( j ) s can be bounded in a way independent of ε, ε and the BRTcovering. By a choice of a sufficiently small ε beforehand, iii) (for ε -version) follows. We have provedthe proposition with ε (in place of the previous ε ). (cid:3) Patching up angular integrals over X ; proof for the simple type. We are going to study themain issue(7.32) Z X Tr a + s ( t, x, x ) dv X ( x ) where we recall (by (5.41)) a + s ( t, x, y ) (= a + s,m ( t, x, y )) (ˆ b + j,s = ˆ b + j,s,m )= 12 π N X j =1 Z π − π e − ˆ hj, +( x,e − iu ◦ y ) t ˆ b + j,s ( x, e − iu ◦ y ) e − imu du, s = n, n − , n − , . . . . (7.33)For this, one would like to patch up those integrals R π J ( j ) s du of the last subsection over j . However, a + s ( t, x, y ) is not canonically defined by our method and is in fact dependent on the choice of BRTcharts. A direct study of it appears inefficient (unless one sticks to a fixed covering of BRT charts).It turns out to be more effective if instead, one studies its equivalence (cf. (5.54) in the asymptoticsense):(7.34) Z X Tr e − t e (cid:3) + b,m ( x, x ) dv X ( x ) in which e − t e (cid:3) + b,m ( x, y ) is of course independent of choice of BRT charts.Suppose a δ > and an s = n, n − , . . . are given. Assume that the BRT covering { ˆ D j } j satisfiesProposition 7.16 in which by using compactness, one can find a (finite) covering { Ω α } α of X , Ω α ⋐ ˆ D j if ˆ D j ∩ Ω α = ∅ , and an ε > such that the conclusion i), ii) and iii) of that proposition hold with eachof these Ω α and this ε . As indicated in Proposition 7.2, whenever necessary, one can shrink the size of Ω α without changing ε . For ρ = ε , we assume (possibly after shrinking Ω α and using compactness)for each α, j , and for some (possibly big) m > ,(7.35) θ -coordinates of Ω α , ˆ D j lie inside of [ − ρ, ρ ] , [ − mρ, mρ ] respectively.Let { g α ( x ) } α be a partition of unity subordinate to this covering (i.e. Supp g α ⋐ Ω α ). We furtherassume each g α is of small support in the sense of Definition 7.7. One sees that as t → + (7.36) Z X g α ( x )Tr e − t e (cid:3) + b,m ( x, x ) dv X ( x ) ∼ X j X s = n,n − , ··· t − s Z π J ( j ) s ( g α ( x )) du where the term to the right is computed with respect to any given BRT covering of X , including but notrestricted to, the previous { ˆ D j } j . Hence at each stage of the computation we may choose convenientBRT charts for the need (as far as the asymptotic expansion as t → + is concerned).By Proposition 7.16, (7.36) is reduced to computing I ( j ) ℓ,s ( g α ) (see (7.30)) (for a fixed g α ). EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 85 Henceforth, in the following we fix an (arbitrarily given) α . As aforementioned, we are free to resetthe BRT charts { ˆ D j } j (with certain cut-off functions). To do so, we make the following definition forconvenience. Definition 7.17. Fix an x ∈ X . { ˆ D j } j ( ˆ D j ⊂ D j etc. notations as in the beginning of this section) a(finite) covering of X , is said to be a covering by distinguished BRT charts at x provided that x ∈ ˆ D j for some j and x ˆ D k for all k = j .Now, we can further assume that for the above fixed α and for an x ∈ Ω α , { ˆ D j } j is distinguished at x in the sense of Definition 7.17. In fact we can assume a little more that Ω α ⋐ ˆ D j and Ω α ∩ ˆ D k = ∅ for k = j ; namely { ˆ D j } j is distinguished at x for each x ∈ Ω α . Also, we assume that (7.35) issatisfied.We shall now choose the cut-off function σ j , in notation of (7.30), that satisfies (see lines above(5.36))(7.37) Z mρ − mρ σ j ( u ) du = Z ρ − ρ σ j ( u ) du = 1 , Supp σ j ⊂ ] − ρ, ρ [ and choose χ j ≡ , so τ j ≡ , on Ω α (see loc. cit. ).With the above setup, some simplifications for (7.36) occur. Firstly,(7.38) Z X g α ( x )Tr e − t e (cid:3) + b,m ( x, x ) dv X ( x ) ∼ X s = n,n − , ··· t − s Z π J ( j ) s ( g α ( x )) du. We are reduced, by Proposition 7.16, to computing the integrals in (7.30).Secondly, in notation of (7.30) there is an angular integral(7.39) Z ε − ε σ j ( θ ( Y ) + u ) du. For a fixed Y ∈ Ω α , θ ( Y ) ∈ [ − ρ, ρ ] by (7.35), and for u going through [ − ε, ε ] = [ − ρ, ρ ] of (7.39),one sees that θ ( Y ) + u covers [ − ρ, ρ ] , it follows from (7.37) that the angular integral (7.39) is .Thirdly, by the above condition on χ j and τ j one obtains, with (7.39) ≡ , the following for (7.30).(7.40) I + , ( j ) ℓ ( γ ℓ ) ,s ( g α ( x )) (= I + , ( j ) X pℓ ( γℓ ) ,s ( g α ( x ))) = 12 π Z X pℓ ( γℓ ) g α ( Y )Tr b + j ,s ( z ( Y ) , z ( Y )) dv X pℓ ( γℓ ) ( Y )( ℓ ≥ s = n, n − , . . . ) .Finally, recall α + s ( x ) as in our main result Theorem 1.3 (cf. Theorem 6.1) defined in (6.1) which isindependent of choice of BRT charts and cut-off functions (cf. Remarks 1.6 and 5.7). Indeed one sees,for x ∈ Ω α ,(7.41) π b + j ,s ( z ( x ) , z ( x )) = α + s ( x ) (= α + s,m ( x )) by (6.1) and the choice of distinguished BRT charts here.In sum, since the above applies to each Ω α in the covering { Ω α } α , by (7.40) and (7.41) we havereached the following invariant expressions (independent of choice of BRT coverings) ( k ≥ ℓ ≥ S + ℓ ( γ ℓ ) ,s ( g α ) (= S + X pℓ ( γℓ ) ,s,m ( g α )) ≡ S + , ( j ) ℓ ( γ ℓ ) ,s ( g α ) = Z X pℓ ( γℓ ) g α ( Y )Tr α + s,m ( Y, Y ) dv X pℓ ( γℓ ) ( Y ) S + ℓ ( γ ℓ ) ,s (= S + X pℓ ( γℓ ) ,s,m ) ≡ X α S + ℓ ( γ ℓ ) ,s ( g α ) = Z X pℓ ( γℓ ) Tr α + s,m ( Y, Y ) dv X pℓ ( γℓ ) ( Y ) . (7.42)Now (7.36) and (7.38) can be given, by using (7.42) and Proposition 7.16, as follows. EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 86 First, we classify the set { Ω α } α by writing(7.43) χ ( α ) (= χ (Ω α )) = ℓ if Ω α T X p ℓ = ∅ and for any ℓ ′ > ℓ , Ω α T X p ℓ ′ = ∅ , ≤ ℓ, ℓ ′ ≤ k. Note d c below (with a specific c ) is given without ambiguity (cf. (7.28)) by the local nature of Ω α and g α . Proposition 7.18. In the notation above and in terms of the functions of (7.42) , let α , δ > and any m ∈ N be given. Then we have, as t → + , for g α with Supp g α ⋐ Ω α such that χ ( α ) = ℓ , Z X g α ( x )Tr e − t e (cid:3) + b,m ( x, x ) dv X ( x ) ∼ X s = n,n − ,... t − s A s ( g α ) where A s ( g α ) is given by A s ( g α ) = (cid:0) p X q =1 e − i πqmp (cid:1) S + X,s ( g α ) + X c ≤ i ( c ) ≤ ℓ ( d c S + i ( c )( γ i ( c ) ) ,s ( g α ) √ t e i ( c )( γi ( c )) + O ( √ t e i ( c )( γi ( c )) +1 ))+ η s ( g α ) (7.44) where i ( c ) = i ( c, g α ) , d c = d c,g α ,m,I , I = S + i ( c )( γ i ( c ) ) ,s ( g α ) by (7.30) and (7.28) , and η s ( g α ) (which equals η ( j ) s ( g α ) in notation of Proposition 7.16 and distinguished BRT charts at x ) satisfies the estimate | η s ( g α ) | ≤ δ · Z X g α dv X for s = n, n − , n − , . . . , − m . In the remaining of this subsection, to streamline the argument we assume the simplest case that i ) each X p ℓ , ≤ ℓ ≤ k , is connected ii ) X = X p ) X p ) X p . . . ) X p k . (We postpone the general case to the next subsection.) One sees p | p | . . . | p k .Hence all types reduce to simple types (cf. Definition 7.10). In this case, the subscript γ ℓ in ℓ ( γ ℓ ) will henceforth be dropped throughout the remaining ofthis subsection .It will take a bit more work to sum (7.44) over α . The numerical factor d c (cf. (7.24)) in thissimplified case satisfies the following. For smooth functions g, g ′ of small support (Definition 7.7), if i ( c, g ) = i ( c, g ′ ) ( ≤ ∞ ) , then d c,g,m = d c,g ′ ,m . It is useful to set, for g = g α with χ ( α ) ≥ ℓ in ii) below( χ ( α ) as in (7.43)), i ) D ,g (= D ,g,m ) ≡ X c, i ( c,g )=1 d c,g,m = p X q =1 e − i πqmp ; ii ) ( k ≥ ℓ ≥ D ℓ,g (= D ℓ,g,m ) ≡ X c, i ( c,g )= ℓ d c,g,m . (7.45)Suppose α, β ∈ S ℓ ′ ≥ ℓ χ − ( ℓ ′ ) . As said above, one sees D ℓ,g α = D ℓ,g β for ≤ ℓ ≤ k (because i ( c, g α ) = ℓ if and only if i ( c, g β ) = ℓ here). We write Definition 7.19. D ℓ (= D ℓ,m ) := D ℓ,g α (= D ℓ,g α ,m ) for any α with χ ( α ) ≥ ℓ , ≤ ℓ ≤ k . EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 87 By using (7.44) of Proposition 7.18 and Definition 7.19, one sees, for α ∈ χ − ( ℓ ) , α ∈ χ − ( ℓ ) , Z X g α ( x )Tr e − t e (cid:3) + b,m ( x, x ) dv X ( x ) ∼ X s = n,n − ,... t − s × η s ( g α ) + D S +1 ,s ( g α ) + X c,i ( c )=2 (cid:0) d c S +2 ,s ( g α ) √ t e + O ( √ t e +1 ) (cid:1) + . . . + X c,i ( c )= ℓ (cid:0) d c S + ℓ,s ( g α ) √ t e ℓ + O ( √ t e ℓ +1 ) (cid:1) = X s t − s (cid:16) η s ( g α ) + D S +1 ,s ( g α ) + (cid:0) D S +2 ,s ( g α ) √ t e + O ( √ t e +1 ) (cid:1) + . . . + (cid:0) D ℓ S + ℓ,s ( g α ) √ t e ℓ + O ( √ t e ℓ +1 ) (cid:1)(cid:17) . (7.46)Combining (7.43) and (7.46) yields the following as t → + (where { α } α = χ − (1) ∪ χ − (2) ∪ χ − (3) . . . ): Z X Tr e − t e (cid:3) + b,m ( x, x ) dv X ( x ) (cid:16) = X α Z X g α ( x )Tr e − t e (cid:3) + b,m ( x, x ) dv X ( x ) (cid:17) ∼ X s = n,n − ,... t − s (cid:16)(cid:0) X α ∈ χ − (1) D S +1 ,s ( g α ) (cid:1) + (cid:0) ( X α ∈ χ − (2) D S +1 ,s ( g α )) + ( X α ∈ χ − (2) D S +2 ,s ( g α ) √ t e + O ( √ t e +1 )) (cid:1) + (cid:0) ( X α ∈ χ − (3) D S +1 ,s ( g α )) + ( X α ∈ χ − (3) D S +2 ,s ( g α ) √ t e + O ( √ t e +1 ))+ ( X α ∈ χ − (3) D S +3 ,s ( g α ) √ t e + O ( √ t e +1 )) (cid:1) + . . . + X α η s ( g α ) (cid:17) . (7.47)We rearrange (7.47) as (only keeping terms in leading order) X s t − s × (cid:16) X α η s ( g α ) + (cid:0) X α ∈ χ − (1) D S +1 ,s ( g α ) + X α ∈ χ − (2) D S +1 ,s ( g α ) + X α ∈ χ − (3) D S +1 ,s ( g α ) + . . . (cid:1) + (cid:0) ( X α ∈ χ − (2) D S +2 ,s ( g α ) √ t e + . . . ) + ( X α ∈ χ − (3) D S +2 ,s ( g α ) √ t e + . . . ) + ( X α ∈ χ − (4) . . . ) + . . . (cid:1) + (cid:0) ( X α ∈ χ − (3) D S +3 ,s ( g α ) √ t e + . . . ) + ( X α ∈ χ − (4) D S +3 ,s ( g α ) √ t e + . . . ) + ( X α ∈ χ − (5) . . . ) + . . . (cid:1) + . . . (cid:17) (7.48)which equals (by (7.42) and S + ℓ,s ( g α ) = 0 for α ∈ χ − ( ℓ ′ ) with ℓ ′ < ℓ ), X s = n,n − ... t − s × (cid:16) X α η s ( g α ) + D S +1 ,s + (cid:0) D S +2 ,s √ t e + O ( √ t e +1 ) (cid:1) + (cid:0) D S +3 ,s √ t e + O ( √ t e +1 ) (cid:1) + . . . (cid:1) + . . . (cid:17) . (7.49)For s = n , we have S + ℓ,n = π (2 π ) − n vol( X p ℓ ) (see iii) of Proposition 7.2).For the term given by the sum P α η s ( g α ) in (7.49), by Proposition 7.18 we obtain P α | η s ( g α ) | ≤ δ · vol( X ) (as P α R X g α dv X = vol( X ) ), s = n, n − , . . . , − m , where δ > and m ∈ N are arbitrarilyprescribed. By the definition of asymptotic expansion (cf. Definition 5.4) and the fact that (7.49) hasbeen an asymptotic expansion, one sees the term P α η s ( g α ) becomes immaterial to the exact form ofthe asymptotic expansion.Further, the asymptotic expansion of R X Tr a + s ( t, x, x ) dv X ( x ) of (7.32) basically follows from that of R X Tr e − t e (cid:3) + b,m ( x, x ) dv X ( x ) .We have now proved (part of) the main result of this section. EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 88 Theorem 7.20. (cf. Theorem 1.14) Suppose X = X p ) X p ) · · · ) X p k = X p k with each stratum X p ℓ a connected submanifold. Let a + s ( t, x, y ) (= a + s,m ( t, x, y )) , s = n, n − , . . . , be as in (5.54) . Write e for the(real) codimension of X p (which is an even number, cf. Remark 7.22 below). (Recall the numerical factors D ℓ,m as given in Definition 7.19 and the integrals S + ℓ,s (= S + ℓ,s,m ) in (7.42) with subscripts simplified inthe present case.) Then the following holds.i) As t → + , Z X Tr e − t e (cid:3) + b,m ( x, x ) dv X ( x ) ∼ D ,m (cid:0) (2 π ) − (2 πt ) − n vol( X ) + t − n +1 S +1 ,n − + t − n +2 S +1 ,n − + . . . (cid:1) + (2 π ) − ( n +1) D ,m vol( X p ) t − n + e + O ( t − n + e ) . (7.50) In particular, by P p q =1 e − πip qm = p for p | m and otherwise, one has D ,m = p if p | m . If p | m (thus p | m too), then D ,m , D ,m > .ii) In the asymptotic expansion (7.50) , all the coefficients of t j for j being half-integral, vanish.iii) As a consequence of (7.49) and ii) (7.51) Z X Tr a + s,m ( t, x, x ) dv X ( x ) ∼ D ,m S +1 ,s + D ,m S +2 ,s t e + O ( t e +1 ) , as t → + . The similar results hold true for the case R X Tr e − t e (cid:3) − b,m ( x, x ) dv X ( x ) and R X Tr a − s,m ( t, x, x )) dv X ( x ) aswell.Proof. It remains to prove ii) of the theorem. Recall the last two paragraphs of the proof of Proposi-tion 7.2, especially the item δ ) there. In the present case, by scaling (ˆ y → √ t ˆ y ) and using (7.14), itreduces to computing the expansion (in √ t ) of a ) ˆ b + j,s (( √ t ˆ y, Y ) , e − iu ◦ ( √ t ˆ y, Y )) b ) dv X ( x ) (7.52)for a fixed u .Write g u ( x ) = e − iu ◦ x and ˆ b + j,s (( √ t ˆ y, Y ) , e − iu ◦ ( √ t ˆ y, Y )) = (ˆ b j,s ◦ (id , g u ))(( √ t ˆ y, Y ) , ( √ t ˆ y, Y )) . By δ )mentioned above, g u (0 , Y ) is only away from (0 , Y ) by a small difference ( ≤ ε ) in their θ -coordinates,hence by continuity, g u (( √ t ˆ y, Y )) lies in an O (2 ε ) -small neighborhood of (0 , Y ) (as t → + ), givingthat the Taylor expansion of ˆ b j,s ( x, y ) , x = ( √ t ˆ y, Y ) , y = e − iu ◦ x , around x = y = (0 , Y ) ≡ Y ≡ canbe done in terms of integral powers of √ t ˆ y i where ˆ y i is in ˆ y . Hence the coefficients of the t j for j beinghalf-integral must involve an odd power of some variable ˆ y i in ˆ y . Since ˆ y sits in an even dimensionalspace (cf. i) of Remark 7.22 below), dv X ( x ) is of integral power in t . With a), b) of (7.52), byusing i) the claim (7.14), ii) R ∞−∞ e − ˆ y i ˆ y ni d ˆ y i = 0 for an odd number n and iii) for a polynomial P ( x ) , R ∞ / √ t e − x P ( x ) dx ∼ O ( t ∞ ) (as t → + ), our assertion about the asymptotic expansion in ii) of thetheorem follows. (cid:3) Remark . One may think of the second line in (7.50) as the main terms which remind one of theclose relation between the Kodaira Laplacian and Kohn Laplacian (cf. Proposition 5.1). However, onekey point in this paper is the idea that if the S action is locally free (but not globally free) on X , thenthis relation cannot be altogether extended to their heat kernels. In this regard the correction terms exist, and consist in the third line of (7.50) linked up with the higher strata of the (locally free) S action beyond the principal stratum. Remark . i) We argue e ℓ is even. X p ℓ is S invariant; T X p ℓ = ( R T ⊥ ∩ T X p ℓ ) ⊕ R T | X pℓ where R T is the line subbundle of T X generated by ∂/∂θ such that R T ⊕ R T ⊥ = T X . In a BRT chart U × ] ε, ε [ we denote U × { } ( ⊂ X ) by ˜ U . Write ( R T ⊥ ∩ T X p ℓ ) | ˜ U ∩ X pℓ ≡ E . For any given p ∈ U , EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 89 with ˜ p = p × { } , one may choose a BRT coordinate such that E ˜ p ⊂ T ˜ p ˜ U (see the proof of Lemma 7.6where T ˜ p ˜ U = R T ⊥ | ˜ p ). Without loss of generality we may assume ˜ p ∈ X p ℓ . Write g = e i π/p ℓ ( ∈ S ) which is CR and an isometry with the fixed point set X p ℓ . By dg ◦ J = J ◦ dg = J on E ˜ p with J thecomplex structure of T ˜ p ˜ U , E ˜ p is invariant under J . It follows E is of even dimension, so X p ℓ is of odddimension, i.e. e ℓ is even. ii) For ≤ ℓ ≤ k we can write X p ℓ → M p ℓ for a complex manifold M p ℓ , asan S fiber bundle. The quantities α ± s ( x ) by the construction (see (7.40)-(7.42)) is S invariant hencedescend to M p ℓ . One sees S ± X ℓ ,s ( ≡ S ± ℓ,s ) = πp ℓ S ± M pℓ,s ( S ± M pℓ,s = R M pℓ Tr α ± s dv M pℓ ) . Here, the metric on M p ℓ (cf. dv M pℓ ) is defined in a way similar to that given in (4.1). This suggests a question of howthe heat kernels (for the locally free S action) of the present paper may be connected with (certainsuitably defined) heat kernels in the orbifold base X/S . In a certain Riemannian setting, some workin a similar direction has been done (cf. [57, Theorems 3.5, 3.6]).7.5. Types for S stratifications; proof for the general type. Lastly, to modify the above reasoningto the case beyond the simple type is essentially not difficult. Suppose, say, X p has several connectedcomponents Y i such that the simple type condition is assumed along each component Y i . Then, clearlythe above argument applies to the individual Y i and the result is just to sum up over i . Withoutassuming the simple type condition on Y i , say, inside some Y i the next stratum X p has seated severalcomponents Z j or some components Z j are seated even outside of each Y i . Then by localizationargument along each Z j just as done above, one repeats the pattern similarly. The process continues.We are now motivated to transplant the notion of “type”, “class” in Definition 7.1 for the integral I of (7.1) into the geometry of the stratification of the S action.For a connected component X p ℓ ( γℓ ) ⊂ X p ℓ , γ ℓ = 1 , . . . , s ℓ , contained in the higher dimensionalconnected components of the strata ( X =) X p i γi ) X p i γi . . . ) X p if ( γif ) ) X p if +1( γif +1) = X p ℓ ( γℓ ) where i = 1 < i < . . . < i f < i f +1 = ℓ ∈ { , , , . . . , ℓ − , ℓ } , we define its type τ ( X p ℓ ( γℓ ) ) by(7.53) τ ( X p ℓ ( γℓ ) ) ≡ τ ( ℓ ( γ ℓ )) := ( i ( γ i ) , i ( γ i ) , . . . , i f ( γ i f ) , i f +1 ( γ i f +1 )) , i = γ i = s = 1; i f +1 = ℓ. One has p i | p i | . . . | p i f +1 (cf. Remark 1.16 for a similar case).The notions such as simple type , class and length l ( τ ) are defined similarly, cf. Definition 7.1. (Nodefinition of trivial type is given here. See iii) of Definition 7.23 below in which i ( c, [ τ ]) = ∞ corre-sponds to the trivial type, cf. iii) of Definition 7.10.)Recall if M ⊂ N is a finite disjoint union of submanifolds M j , then the dimension of M is max j { dim R M j } and the codimension of M is dim R N − dim R M .The following definition, which is bit tedious yet bears a great similarity as previously, is set up forthe immediate use in the general situation. Definition 7.23. i) Write ν ℓ = { [ τ ]; τ = τ ( X p ℓ ( γℓ ) ) , γ ℓ = 1 , , . . . , s ℓ } for the set of equivalence classesof types τ = τ ( X p ℓ ( γℓ ) ) of connected components X p ℓ ( γℓ ) in X p ℓ .ii) Write (similar to (7.42), (7.41)) S + ℓ ( γ ℓ ) ,s (= S + X pℓ ( γℓ ) ,s,m ) = Z X pℓ ( γℓ ) Tr α + s ( Y, Y ) dv X pℓ ( γℓ ) ( Y ) ( α + s = α + s,m ) associated with X p ℓ ( γℓ ) .iii) Let [ τ ] = [( i ( γ i ) , i ( γ i ) , . . . , i f ( γ i f ) , i f +1 ( γ i f +1 ))] be given. If c | p , define i ( c, [ τ ]) = 1 . (Henceit is independent of [ τ ] .) If c p and c | p ℓ , ℓ = i s for some s , ≤ s ≤ f + 1 , such that c i s ′ for all s ′ < s . Then i ( c, [ τ ]) := ℓ ≥ . If c p i s for ≤ s ≤ t + 1 , i ( c, [ τ ]) := ∞ . We may write i ( c ) for i ( c, [ τ ]) .iv) For i ( c, [ τ ]) ≥ , define the numerical factors d c,m, [ τ ] correspondingly as in (7.28). For i ( c ) = 1 ,define d c,m, [ τ ] = d c (which is independent of τ ) as in (7.24). EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 90 v) For a given [ τ ] , if i ( c ) = 1 , then define the weight factors D as in (7.45) (which is independentof τ ) and if i ( c ) = 1 , ∞ , define D ℓ, [ τ ] (= D ℓ,m, [ τ ] ) ≡ P c, i ( c )= ℓ d c,m, [ τ ] .vi) Write e i q , [ τ ] ≡ e i q ( γ iq ) with τ ( ℓ ( γ ℓ )) = ( i ( γ i ) , . . . , i q ( γ i q ) , . . . , i f +1 ( γ i f +1 )) for the codimensionof X p iq ( γ iq ) . Obviously, e i , [ τ ] < e i , [ τ ] < · · · . For [ τ ] ∈ ν ℓ , write e [ τ ] ≡ e i f +1 ( γ if +1 ) , i.e. e ℓ ( γ ℓ ) .vii) Write e = min [ τ ] ∈ ν ℓ ≤ ℓ ≤ k e [ τ ] (= min τ,l ( τ )=2 e [ τ ] by vi) above ) and for ℓ ≥ , ˆ ν ℓ = { [ˆ τ ℓ ] ∈ ν ℓ ; ˆ τ ℓ = (1 , ℓ ( γ ℓ )) of length two such that e ℓ ( γ ℓ ) = e , i.e. e [ˆ τ ℓ ] = e } ⊂ ν ℓ . Of course, it is not ruled out that for some values of ℓ , ˆ ν ℓ could be an empty set. Intuitively, one thinksof e as the minimal codimension among those connected components X p ℓ ( γℓ ) such that if X p ℓ ′ ) X p ℓ ( γℓ ) , then X p ℓ ′ = X p the principal stratum.viii) For a fixed [ κ ] ∈ ν ℓ in i), write (see (7.53) for τ ( ℓ ( γ ℓ )) ≡ τ ( X p ℓ ( γℓ ) ) ) Z [ κ ] ,s = X γ ℓ , [ τ ( ℓ ( γ ℓ ))]=[ κ ]1 ≤ γ ℓ ≤ s ℓ S + ℓ ( γ ℓ ) ,s . For the case of general type, we can obtain results corresponding to (7.47), (7.48) and (7.49) (yetcomplicated in expressions here). We are content with summarizing the final result as follows. Z X Tr e − t e (cid:3) + b,m ( x, x ) dv X ( x ) ∼ X s = n,n − ... t − s × (cid:16) D S +1 ,s + X [ κ ] ∈ ν (cid:0) D , [ κ ] Z [ κ ] ,s √ t e [ κ + O ( √ t e [ κ +1 ) (cid:1) + X [ κ ] ∈ ν (cid:0) D , [ κ ] Z [ κ ] ,s √ t e [ κ + O ( √ t e [ κ +1 ) (cid:1) + . . . (cid:17) . (7.54)The following main result of this subsection parallels Theorem 7.20 in the last subsection. Bycomparison, to collect the coefficients for the next leading order in t or √ t in (7.54) here, we havea slightly more complicated summation (regarded as part of corrections as indicated in Remark 7.21)in the formula below. Note that the conversion to the stated form of Theorem 1.14 is nothing but adirect consequence of an examination (slightly tedious) of the various definitions here. Theorem 7.24. (cf. Theorem 1.14) Notations as in Theorem 7.20 without assuming the conditions ofconnectedness and simple-type there. The weight factors D ℓ,m, [ τ ] , the integrals S + ℓ ( γ ℓ ) ,s (= S + ℓ ( γ ℓ ) ,s,m ) , e , ˆ τ ℓ etc. are just given above. One has the following.i) As t → + , Z X Tr e − t e (cid:3) + b,m ( x, x ) dv X ( x ) ∼ D ,m (cid:0) (2 π ) − (2 πt ) − n vol( X ) + t − n +1 S +1 ,n − + t − n +2 S +1 ,n − + . . . (cid:1) + t − n + e ( X [ˆ τ ℓ ] ∈ ˆ ν ℓ ≤ ℓ ≤ k D ℓ,m, [ˆ τ ℓ ] Z [ˆ τ ℓ ] ,n ) + O ( t − n + e +12 ) (7.55) (where recall Z [ˆ τ ℓ ] ,n = (2 π ) − ( n +1) P γ ℓ , ≤ γ ℓ ≤ s ℓ codim X pℓ ( γℓ ) = e vol( X p ℓ ( γℓ ) ) > in the locally free case of the S action). If p ℓ | m (thus p | m too), then D ,m , D ℓ,m, [ˆ τ ℓ ] > .ii) In the asymptotic expansion (7.55) , all the coefficients of t j for j being half-integral, vanish.iii) As a consequence of (7.54) and ii), (7.56) Z X Tr a + s,m ( t, x, x ) dv X ( x ) ∼ D ,m S +1 ,s + t e ( X [ˆ τ ℓ ] ∈ ˆ ν ℓ ≤ ℓ ≤ k D ℓ,m, [ˆ τ ℓ ] Z [ˆ τ ℓ ] ,s ) + O ( t e +1 ) EAT KERNEL ASYMPTOTICS, LOCAL INDEX THEOREM AND TRACE INTEGRALS FOR CR MANIFOLDS WITH S ACTION 91 (where Z [ˆ τ ℓ ] ,s = P γ ℓ , ≤ γ ℓ ≤ s ℓ codim X pℓ ( γℓ ) = e S + ℓ ( γ ℓ ) ,s ).The similar results hold for R X Tr e − t e (cid:3) − b,m ( x, x )) dv X ( x ) and R X Tr a − s,m ( t, x, x ) dv X ( x ) .Remark . The quantities involved above are computable in the sense that they are basically re-duced to those involved in the (ordinary) Kodaira heat kernel by ii) of Definition 7.23, cf. Remark 1.9. Remark . It is not obvious how one can compute the supertrace integral, hence our index theo-rem 6.4, Corollary 6.5 solely by techniques similar to those derived in (7.55), partly because here weare not using the off-diagonal estimate of Theorem 5.9 which is partly based on a cancellation result inBerenzin integral (see the proof of that theorem). These results (of estimate and cancellation) appearto lie beyond what the geometry of the stratifications can reveal as done in this Section 7. Acknowledgements. The first named author would like to thank the Ministry of Science and Tech-nology of Taiwan for the support of the project: 104-2115-M-001-011-MY2. 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