Action of micro-differential operators on quantized contact transformations
aa r X i v : . [ m a t h . AG ] F e b Action of micro-differential operators on quantizedcontact transformations
Mehdi BenchoufiFebruary 19, 2021
Abstract
Quantized contact transforms (QCT) have been constructed in [SKK]. Wegive here a complete proof of the fact that such QCT commute with the action ofmicrodifferential operators. To our knowledge, such a proof did not exist in theliterature. We apply this result to the microlocal Radon transform.
Contents O -modules and D -modules . . . . . . . . . . . . . . . . . . . . 82.4 E -modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Hyperfunctions and microfunctions . . . . . . . . . . . . . . . 92.6 Integral transforms for sheaves and D -modules . . . . . . . . . 92.7 Microlocal integral transforms . . . . . . . . . . . . . . . . . . 102.8 Complements on the functor µhom . . . . . . . . . . . . . . . 123 Complex quantized contact transformations . . . . . . . . . . . . . . . 143.1 Kernels on complex manifolds . . . . . . . . . . . . . . . . . . 143.2 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 Radon transform for sheaves . . . . . . . . . . . . . . . . . . . . . . . 224.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Projective duality: geometry . . . . . . . . . . . . . . . . . . . 234.3 Projective duality for microdifferential operators . . . . . . . . 254.4 Projective duality for microfunctions . . . . . . . . . . . . . . 274.5 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 Introduction
For a manifold M , let us denote by T ∗ M the cotangent bundle and • T ∗ M the bundle T ∗ M with the zero section removed. We will consider the following situation: let X and Y be two complex manifolds of the same dimension, a closed submanifold Z of X × Y , open subset U ⊂ • T ∗ X and V ⊂ • T ∗ Y and assume that the conormal bundle • T ∗ Z ( X × Y ) induces a contact transformation • T ∗ Z ( X × Y ) ∩ ( U × V a ) ∼ u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ ∼ ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ • T ∗ X ⊃ U ∼ / / V ⊂ • T ∗ Y. (1.1)Let F ∈ D b ( X ) and let ϕ K ( F ) denote the contact transformation of F with kernel K ∈ D b ( X × Y ). We will prove an isomorphism between µhom ( F, O X ) on U ∩ T ∗ M X and µhom ( ϕ K ( F ) , O Y ) on V ∩ T ∗ N Y , which follows immediately from [KS90, Lem.11.4.3]. Our main result will be the commutation of this isomorphism to the actionof microdifferential operators. Although considered as well-known, the proof of thiscommutation does not appear clearly in the literature (see [SKK, p. 467]), and is farfrom being obvious. In fact, we will consider a more general setting, replacing sheavesof microfunctions with sheaves of the type µhom ( F, O X ). Moreover, assume that X and Y are complexification of real analytic manifolds M and N respectively. Then,it is known that, under suitable hypotheses, one can quantize this contact transformand get an isomorphism between microfunctions on U ∩ T ∗ M X and microfunctions on V ∩ T ∗ N Y [KKK86].Then, we will specialize our results to the case of projective duality. We will studythe microlocal Radon transform understood as a quantization of projective duality, bothin the real and the complex case.In the real case, denote by P the real projective space (say of dimension n ), by P ∗ its dual and by S the incidence relation: S := { ( x, ξ ) ∈ P × P ∗ ; h x, ξ i = 0 } . (1.2)In this setting, there is a well-known correspondance between distributions on P and P ∗ due to Gelfand, Gindikin, Graev [GGG82] and to Helgason [Hel80]. However, it isknown since the 70s under the influence of the Sato’s school, that to well-understandwhat happens on real (analytic) manifolds, it may be worth to look at their complexi-fication.Hence, denote by P the complex projective space of dimension n , by P ∗ the dualprojective space and by S ⊂ P × P ∗ the incidence relation. We have the correspondence • T ∗ S ( P × P ∗ ) ∼ y y ssssssssss ∼ % % ❑❑❑❑❑❑❑❑❑❑ • T ∗ P ∼ / / • T ∗ P ∗ (1.3) 2his contact transformation induces an equivalence of categories between perversesheaves modulo constant ones on the complex projective space and perverse sheavesmodulo constant ones on its dual, as shown by Brylinski [Bry86], or between coherent D -modules modulo flat connections, as shown by D’Agnolo-Schapira [DS94].In continuation of the previous cited works, we shall consider the contact transforminduced by (1.3) • T ∗ S ( P × P ∗ ) ∩ ( • T ∗ P P × • T ∗ P ∗ P ∗ ) ∼ u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ ∼ ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ • T ∗ P P ∼ / / • T ∗ P ∗ P ∗ (1.4)The above contact transformation leads to the well-known fact that the Radon trans-form establishes an isomorphism of sheaves of microfunctions on P and P ∗ (see [KKK86]).We will apply our main result to prove the commutation of this isomorphism to theaction of microdifferential operators. We will use the langage of sheaves and D -modules and we refer the reader to [KS90] and[Kas03] for a detailed developement of these topics. We will denote by k a commutativeunital ring of finite global dimension. Notations for integral transforms
Let
X, Y be real manifolds and S a closedsubmanifold X × Y . Consider the diagrams X f ←− S g −→ Y , X q ←− X × Y q −→ Y . Let F ∈ D b ( k Y ) and K ∈ D b ( k X × Y ). The integral transform of F with respect to thekernel K is defined to be Φ K ( F ) := R q ( K ⊗ q − F ). We will denote Φ S ( F ) the integraltransform of F with respect to the kernel k S [ d S − d X ]. Results on the functor µhom
To establish our main results, we will need thefollowing complement on the functor µhom .For ( M i ) i =1 , , , three manifolds, we write M ij := M i × M j (1 ≤ i, j ≤ ◦ : D b ( k M ) × D b ( k M ) −→ D b ( k M )( K , K ) K ◦ K := R q ( q − K ⊗ q − K ) ≃ R q δ − ( K ⊠ K ) . (1.5)We add a subscript a to p j to denote by p aj the composition of p j and the antipodalmap on T ∗ M j . We define the composition of kernels on cotangent bundles (see [KS90,Prop. 4.4.11]) a ◦ : D b ( k T ∗ M ) × D b ( k T ∗ M ) −→ D b ( k T ∗ M )( K , K ) K a ◦ K := R p ( p − a K ⊗ p − K )(1.6) 3et F i , G i , H i respectively in D b ( k M ) , D b ( k M ) , D b ( k M ), i = 1 ,
2. Let U i be an opensubset of T ∗ M ij ( i = 1 , j = i + 1) and set U = U i a ◦ U j = p ( p − a ( U ) ∩ p − ( U ))In [KS90], a canonical morphism in D b ( k T ∗ M ) is constructed µhom ( F , F ) | U a ◦ µhom ( G , G ) | U −→ µhom ( F ◦ G , F ◦ G ) | U . (1.7)We will see that the composition a ◦ is associative and we will see also that the morphism(1.7) is compatible with associativity with respect to a ◦ . Complex contact transformations
Consider now two complex manifolds X and Y of the same dimension n , open C × -conic subsets U and V of • T X and • T Y , respectively,Λ a smooth closed submanifold of U × V a and assume that the projections p | Λ and p a | Λ induce isomorphisms, hence a homogeneous symplectic isomorphism χ : U ∼−→ V :Λ ⊂ U × V ap ∼ w w ♥♥♥♥♥♥♥♥♥♥ p a ∼ ' ' PPPPPPPPPP • T ∗ X ⊃ U ∼ χ / / V ⊂ • T ∗ Y Let us consider a perverse sheaf L on X × Y satisfying ( p − ( U ) ∪ p a − ( V )) ∩ SS( L ) ⊂ Λand a section s of µhom ( L, Ω X × Y/X ) on Λ, where Ω X × Y/X := O X × Y ⊗ q − O Y q − Ω Y .Recall that one denotes by E R X sheaf of rings E R X := µhom ( C ∆ X , Ω X × X/X )[ d X ], and E X the subsheaf of E R X of finite order microdifferential operators. In the followingtheorem, the first statement ( i ) is well-known, see [SKK], ( ii ) is proved in [KS90], thefact that isomorphism (1.8) is compatible with the action of microdifferential operatorswas done at the germ level in [KS90], but from a global perspective, it was announcedfor microfunctions in various papers but no detailed proof exists to our knowledge. Wewill prove our main theorem: Theorem 1.1.
Let G ∈ D b ( C Y ) and assume to be given a section s of µhom ( L, Ω X × Y/X ) ,non-degenerate on Λ . (i) For W ⊂ U , P ∈ E X ( W ) , there is a unique Q ∈ E Y ( χ ( W )) satisfying P · s = s · Q ( P, Q considered as sections of E X × Y ). The morphism induced by sχ − E Y | V −→ E X | U P Q is a ring isomorphism. We have the following isomorphism in D b ( C U ) χ − µhom ( G, O Y ) | V ∼−→ µhom (Φ L [ n ] ( G ) , O X ) | U (1.8)(iii) The isomorphism (1.8) is compatible with the action of E Y and E X on the left andright side of (1.8) respectively. We will see that the action of microdifferential operators in Theorem 1.1 (iii) isderived from the morphism (1.7).
Projective duality for microfunctions
For M a real analytic manifold and X its complexification, we might be led to identify T ∗ M X with i · T ∗ M . We denote by A M , B M , C M the sheaves of real analytic functions, hyperfunctions, microfunctions,respectively.In this article, we will quantize the contact transform associated with the Lagrangiansubmanifold • T ∗ S ( P × P ∗ ). We will construct and denote by χ the homogeneous symplecticisomorphism between • T ∗ P and • T ∗ P ∗ .For ε ∈ Z / Z , we denote by C P ( ε ) the two locally constant sheaf of rank one on P (see Section 4.1 for a precise definition).Let an integer p ∈ Z , ε ∈ Z / Z , we will define the sheaves of real analytic functions A P ( ε, p ), hyperfunctions B P ( ε, p ), microfunctions C P ( ε, p ), on P resp. P ∗ twisted bysome power of the tautological line bundle.For X, Y either the manifold P or P ∗ , for any two integers p, q , we note O X × Y ( p, q )the line bundle on X × Y with homogenity p in the X variable and q in the Y variable.Setting Ω X × Y/X ( p, q ) := Ω X × Y/X ⊗ O X × Y O X × Y ( p, q ), E R X ( p, q ) := µhom ( C ∆ X , Ω X × X/X ( p, q ))[ d X ]and we define accordingly E X ( p, q ). Let us notice that E R X ( − p, p ) is a sheaf of rings.Let n be the dimension of P , (of course n = d P ). For an integer k and ε ∈ Z / Z ,we note k ∗ := − n − − k , ε ∗ := − n − − ε mod (2). We have: Theorem 1.2. (i)
Let k be an integer such that − n − < k < and let s be a globalnon-degenerate section on • T ∗ S ( P × P ∗ ) of H ( µhom ( C S , Ω P × P ∗ / P ∗ ( − k, k ∗ ))) . For P ∈ E P ( − k, k ) , there is a unique Q ∈ E P ∗ ( − k ∗ , k ∗ ) satisfying P · s = s · Q . Themorphism induced by s χ ∗ E P ( − k, k ) −→ E P ∗ ( − k ∗ , k ∗ ) P Q is a ring isomorphism. (ii) There exists such a non-degenerate section s . In fact, we will see that the non-degenerate section of Theorem 1.2 is provided bythe Leray section.Now, from classical adjunction formulas for E -modules, we get a correspondancebetween solutions of systems of microdifferential equations on the projective space andsolutions of systems of microdifferential equations on its dual. We will prove the fol-lowing theorem, which was proved in [DS96] for D -modules,5 heorem 1.3. Let k be an integer such that − n − < k < . Let N be a coherent E P ( − k, k ) -module and F ∈ D b ( P ) . Then, we have an isomorphism in D b ( C • T ∗ P ∗ ) χ ∗ R H om E P ( − k,k ) ( N , µhom ( F, O P ( k ))) ≃ R H om E P ∗ ( − k ∗ ,k ∗ ) (Φ µ S ( N ) , µhom ((Φ C S [ − F, O P ∗ ( k ∗ )))where Φ µ S it is the counterpart of Φ S for E -modules, and will be defined in Section2.7.1.Let us mention that, through a difficult result from [Kas+06], µhom ( F, O P ) is well-defined in the derived category of E -modules. Corollary 1.4.
Let k be an integer such that − n − < k < and ε ∈ Z / Z . Thesection s of theorem 1.2 defines an isomorphism: χ ∗ C P ( ε, k ) | • T ∗ P P ≃ C P ∗ ( ε ∗ , k ∗ ) | • T ∗ P ∗ P ∗ Moreover, this morphism is compatible with the respective action of χ ∗ E P ( − k, k ) and E P ∗ ( − k ∗ , k ∗ ) . Acknowledgements
I would like to express my gratitude to Pierre Schapira forsuggesting me this problem and for his enlightening insights to which this work owesmuch.
In this section, we recall classical results of Algebraic Analysis, with the exception ofsection 2.8. (i) Let M i ( i = 1 , ,
3) be manifolds. For short, we write M ij := M i × M j (1 ≤ i, j ≤ M = M × M × M , M = M × M × M × M , etc.(ii) δ M i : M i −→ M i × M i denote the diagonal embedding, and ∆ M i the diagonal set of M i × M i .(iii) We will often write for short k i instead of k M i and k ∆ i instead of k ∆ Mi andsimilarly with ω M i , etc., and with the index i replaced with several indices ij , etc.(iv) We denote by π i , π ij , etc. the projection T ∗ M i −→ M i , T ∗ M ij −→ M ij , etc.(v) For a fiber bundle E −→ M , we denote by ˙ E −→ M the fiber bundle with thezero-section removed.(vi) We denote by q i the projection M ij −→ M i or the projection M −→ M i andby q ij the projection M −→ M ij . Similarly, we denote by p i the projection T ∗ M ij −→ T ∗ M i or the projection T ∗ M −→ T ∗ M i and by p ij the projection T ∗ M −→ T ∗ M ij . 6vii) We also need to introduce the maps p j a or p ij a , the composition of p j or p ij andthe antipodal map a on T ∗ M j . For example, p a (( x , x , x ; ξ , ξ , ξ )) = ( x , x ; ξ , − ξ ) . (viii) We let δ : M −→ M be the natural diagonal embedding. We follow the notations of [KS90].Let X be a good topological space, i.e. separated, locally compact, countable atinfinity, of finite global cohomological dimension and let k be a commutative unital ringof finite global dimension.For a locally closed subset Z of X , we denote by k Z , the sheaf, constant on Z withstalk k , and 0 elsewhere.We denote by D b ( k X ) the bounded derived category of the category of sheaves of k -modules on X . If R is a sheaf of rings, we denote by D b ( R ) the bounded derivedcategory of the category of left R -modules.Let Y be a good topological space and f a morphism Y −→ X . We denote byR f ∗ , f − , R f ! , f ! , R H om , L ⊗ the six Grothendieck operations. We denote by ⊠ the ex-terior tensor product.We denote by ω X the dualizing complex on X , by ω ⊗− X the sheaf-inverse of ω X andby ω Y/X the relative dualizing complex.In the following, we assume that X is a real manifold. Recall that ω X ≃ or X [dim X ]where or X is the orientation sheaf and dim X is the dimension of X . We denote byD X ( • ),D ′ X ( • ) the duality functor D X ( • ) = R H om ( • , ω X ) , D ′ X ( • ) = R H om ( • , k X ),respectively.For F ∈ D b ( k X ), we denote by SS ( F ) its singular support, also called micro-support. For a a subset Z ⊂ T ∗ X , we denote by D b ( k X ; Z ) the localization of thecategory D b ( k X ) by the full subcategory of objects whose micro-support is containedin T ∗ X \ Z .For a closed submanifold M of X , we denote by ν M , µ M , µhom , the functor ofspecialization, microlocalization along M and the functor of microcalization of R H om respectively.Let M i ( i = 1 , ,
3) be manifolds. We shall consider the operations of compositionof kernels: ◦ : D b ( k M ) × D b ( k M ) −→ D b ( k M )( K , K ) K ◦ K := R q ( q − K ⊗ q − K ) ≃ R q δ − ( K ⊠ K )(2.1) ◦ : D b ( k M ) × D b ( k M ) × D b ( k M ) −→ D b ( k M )( K , K , K ) K ◦ K ◦ K := R q ( q − K ⊗ q − K ⊗ q − K )(2.2) 7et us mention a variant of ◦ : ∗ : D b ( k M ) × D b ( k M ) −→ D b ( k M )( K , K ) K ∗ K := R q ∗ (cid:0) q − ω ⊗ δ !2 ( K ⊠ K ) (cid:1) There is a natural morphism K ◦ K −→ K ∗ K .We refer the reader to [KS90] for a detailed presentation of sheaves on manifolds. O -modules and D -modules We refer to [Kas03] for the notations and the main results of this section.Let ( X, O X ) be a complex manifold. We denote by d X its complex dimension andby D X the sheaf of rings of finite order holomorphic differential operators on X .For an invertible O X -module F , we denote by F ⊗− := H om O X ( F , O X ), the inverseof F . Denote by Mod( D X ) the abelian category of left D X -modules and Mod( D opX ) ofright D X -modules. We denote by Ω X the right D X -module of holomorphic d X forms.Let D b ( D X ) be the bounded derived category of the category of left D X -modules,D bcoh ( D X ) its full triangulated subcategory whose objects have coherent cohomology.Let D bgood ( D X ) be the triangulated subcategory of D b ( D X ), whose objects have allcohomologies consisting in good D X -modules (see [Kas03] for a classical reference).We refer in the following to [Kas03]. Let f : Y −→ X be a morphism of complexmanifolds. We denote by D Y −→ X and D X ←− Y the transfer bimodules.For M ∈ D b ( D X ), N ∈ D b ( D Y ), we denote by f − M , f ∗ N , the pull-back and thedirect image of D -modules respectively. We refer to [DS96] for functorial properties ofinverse and direct image of D -modules. E -modules We refer in the following to [SKK] (see also [Sch85] for an exposition). For a complexmanifold X , one denotes by E X the sheaf of filtered ring of finite order holomorphicmicrodifferential operators on T ∗ X . We denote by D bcoh ( E X ) the full triangulated sub-category of D b ( E X ) whose objects have coherent cohomology.For m ∈ Z , we denote by E X ( m ) the abelian subgroup of E X of microdifferentialoperators of order less or equal to m . For a section P of E X , we denote by σ ( P ) theprincipal symbol of P .Let π X denote the natural projection T ∗ X −→ X . Let us recall that E X is flat over π − ( D X ). To a D X -module M , we associate an E X -module defined by E M := E X ⊗ π − X D X π − X M To a morphism of manifolds f : Y −→ X , we associate the diagram of natural morphisms: T ∗ Y π Y & & ▼▼▼▼▼▼▼▼▼▼▼▼ Y × X T ∗ X f d o o f π / / π (cid:15) (cid:15) T ∗ X π X (cid:15) (cid:15) Y f / / X (2.3) 8here f d is the transposed of the tangent map T f : T Y −→ Y × X T X .For M , N objects of respectively D b ( E X ) and D b ( E Y ), we denote by f − M and f ∗ N the pull-back and the direct image of E -modules respectively. Let M be a real analytic manifold and X a complexification of M . We might be led toidentify T ∗ M X with i · T ∗ M . We denote by A M := O X | M , B M := R H om (D ′ X C M , O X ), C M := µhom (D ′ X C M , O X ), the sheaves of real analytic functions, hyperfunctions, mi-crofunctions, respectively. Let us denote by sp , the isomorphism sp : B M ∼−→ R π M ∗ C M (2.4)There is a natural action of the sheaf of microdifferential operators E X on C M .If Z is a closed complex submanifold of X of codimension d , we note B Z | X := H d [ Z ] ( O X )the algebraic cohomology of O X with support in Z . D -modules Let X and Y be complex manifolds of respective dimension d X , d Y . Let S be a closedsubmanifold X × Y of dimension d S . We set d S/X := d S − d X . Consider the diagramof complex manifolds S f (cid:0) (cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) g (cid:30) (cid:30) ❃❃❃❃❃❃❃❃ e S g (cid:0) (cid:0) ✁✁✁✁✁✁✁✁ f (cid:30) (cid:30) ❃❃❃❃❃❃❃❃ X Y, Y X (2.5)where the second diagram is obtained by interchanging X and Y .Let F ∈ D b ( C X ), G ∈ D b ( C Y ), we defineΦ S ( F ) := R g ! f − F [ d S/Y ] , Φ e S ( G ) := R f ! g − G [ d S/X ]Ψ S ( F ) := R g ∗ f ! F [ d X/S ] , Ψ e S ( G ) := R f ∗ g ! G [ d Y/S ]For K ∈ D b ( C X × Y ), and given the diagram X q ←− X × Y q −→ Y , we define theintegral transform of F with kernel K Φ K ( F ) := R q ( K ⊗ q − F )9 .6.2 Integral transforms for D -modules Let
X, Y be complex manifolds of equal dimension n >
0, and S a complex manifold.Consider again the situation (2.5).We suppose (cid:26) f, g are smooth and proper, S is a complex submanifold of X × Y of codimension c > M ∈ D b ( D X ), N ∈ D b ( D Y ). Let us denote by e S the image of S by the map r : X × Y −→ Y × X, ( x, y ) ( y, x ). One setsΦ S ( M ) := g ∗ f − M , Φ e S ( N ) := f ∗ g − N We refer to [DS94, Prop. 2.6.] for adjonction formulae related to these integraltransforms.Let us recall that we denote by Ω X the sheaf of holomorphic n -forms and let B ( n, S | X × Y := q − Ω X ⊗ q − O X B S | X × Y This ( D Y , D X )-bimodule allows the computation of Φ S because of the isomorphism,proven in [DS94, Prop 2.12] D Y ←− S L ⊗ D S D S −→ X ∼−→ B ( n, S | X × Y leading to Φ S ( M ) ≃ R q ( B ( n, S | X × Y L ⊗ q − D X q − M ) E -modules Let
X, Y be complex manifolds and S is a closed submanifold of X × Y . We consideragain the diagram (2.5) under the hypothesis (2.6).We define the functorD b ( E X ) −→ D b ( E Y ), Φ µS ( M ) := g ∗ f − M We define the E X × Y -module attached to B S | X × Y , C S | X × Y := E B S | X × Y and we consider the ( E Y , E X )-bimodule C ( n, S | X × Y := π − q − Ω X L ⊗ π − q − O X C S | X × Y (2.7) 10ne can notice that E Y ←− S L ⊗ E S E S −→ X ∼−→ C ( n, S | X × Y and hence, we have Φ µS ( M ) ≃ R p a ( C ( n, S | X × Y L ⊗ p − E X p − M )(2.8)Let M ∈ D bgood ( D X ). The functors Φ µS and Φ S are linked through the followingisomorphism in D b ( C • T ∗ Y ), (see [SS94]) E (Φ S ( M )) ≃ Φ µS ( E M )(2.9) Consider two open subsets U and V of T ∗ X and T ∗ Y , respectively and Λ a closedcomplex Lagrangian submanifold of U × V a U × V ap w w ♣♣♣♣♣♣♣♣♣♣♣ p a ' ' ◆◆◆◆◆◆◆◆◆◆◆ T ∗ X ⊃ U V ⊂ T ∗ Y (2.10)As detailed in Section 11.4 of [KS90], let K ∈ D b ( C X × Y ), SS ( K ) its micro-supportand let us suppose that p | Λ , p a | Λ are isomorphisms, K is cohomologically constructiblesimple with shift 0 along Λ and that ( p − ( U ) ∪ p − ( V )) ∩ SS ( K ) ⊂ Λ.Let p = ( p X , p aY ) ∈ Λ and let us consider some section s ∈ H ( µhom ( K, Ω X × Y/Y )) p ,where Ω X × Y/Y := O X × Y ⊗ q − O X q − Ω X . The section s gives a morphism K −→ Ω X × Y/Y in D b ( C X × Y ; p ). Then, there is a natural morphismΦ K [ d X ] ( O X ) −→ O Y (2.11)We recall the result: Theorem 2.1 ([KS90, Th.11.4.9]) . There exists s ∈ H ( µhom ( K, Ω X × Y/Y )) p suchthat the associated morphism Φ K [ d X ] ( O X ) −→ O Y is an isomorphism in the category D b ( C Y ; p Y ) . Moreover, this morphism is compatible with the action of microdifferentialoperators on O X in D b ( C X ; p X ) and the action of microdifferential operators on O Y in D b ( C Y ; p Y )Also, we will make use of the following theorem proven in [KS90, Th. 7.2.1]: Theorem 2.2 ([KS90, Th. 7.2.1]) . Let K ∈ D b ( X × Y ) and assume that(i) K is cohomologically constructible(ii) ( p − ( U ) ∪ ( p a ) − ( V )) ∩ SS ( K ) ⊂ Λ (iii) the natural morphism C Λ −→ µ hom ( K, K ) | Λ is an isomorphism.Then for any F , F ∈ D b ( X ; U ) , the natural morphism χ ∗ µ hom ( F , F ) −→ µ hom (Φ K ( F ) , Φ K ( F )) is an isomorphism in D b ( Y ; V ) . .8 Complements on the functor µhom The next result is well-known although no proof is written down in the literature, toour knowledge.
Lemma 2.3.
Let M , M , M be real manifolds, and K, L, M be objects respectivelyof D b ( k M ) , D b ( k M ) , D b ( k M ) , then the composition of kernels ◦ defined in 2.1 isassociative. We have the following isomorphism ( K ◦ L ) ◦ M ∼−→ K ◦ ( L ◦ M )(2.12) such that for any N ∈ D b ( k M ) , the diagram below commutes: (( K ◦ L ) ◦ M ) ◦ N / / (cid:15) (cid:15) ( K ◦ L ) ◦ ( M ◦ N ) (cid:15) (cid:15) ( K ◦ ( L ◦ M )) ◦ N (cid:15) (cid:15) K ◦ (( L ◦ M ) ◦ N ) / / K ◦ ( L ◦ ( M ◦ N )) . (2.13) Proof.
Consider the following diagram M q (cid:1) (cid:1) ✄✄✄✄✄✄✄✄✄✄ (cid:1) (cid:1) ✄✄✄✄✄✄✄✄✄✄ q (cid:15) (cid:15) q (cid:29) (cid:29) ❀❀❀❀❀❀❀❀❀❀ (cid:29) (cid:29) ❀❀❀❀❀❀❀❀❀❀ (cid:12) (cid:12) (cid:12) (cid:12) (cid:18) (cid:18) (cid:18) (cid:18) M (cid:29) (cid:29) ❀❀❀❀❀❀❀❀❀❀ q (cid:29) (cid:29) ❀❀❀❀❀❀❀❀❀❀ q q (cid:13) (cid:13) M q (cid:1) (cid:1) ✄✄✄✄✄✄✄✄✄✄ (cid:1) (cid:1) ✄✄✄✄✄✄✄✄✄✄ q ~ ~ ~ ~ q (cid:17) (cid:17) M q (cid:3) (cid:3) ✝✝✝✝✝✝✝✝✝✝ q (cid:15) (cid:15) (cid:15) (cid:15) q ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ M M q v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ q (cid:15) (cid:15) (cid:15) (cid:15) q (cid:27) (cid:27) ✽✽✽✽✽✽✽✽✽✽ M M M M M where the thick squares are cartesian, and where for clarity we enforced the notation:the projection M ijk −→ M ij by q kij (independently of order of appearence of the indices),and the projection M ijkl −→ M ij by q klij . We now have:R q ( q − (R q ( q − K ⊗ q − L )) ⊗ q − M ) ≃ R q (R q ( q − K ⊗ q − L ) ⊗ q − M ) ≃ R q ( q − K ⊗ q − L ⊗ q − M ):= K ◦ K ◦ K K ◦ K ◦ K ≃ R q ( q − K ⊗ ( q − (R q ( q − L ⊗ q − M ))))which proves the isomorphism (2.12). And, it follows immediately that given N ∈ D b ( k M ), the diagram (2.13) commutes. µhom We define the composition of kernels on cotangent bundles (see [KS90, section 3.6,(3.6.2)]). a ◦ : D b ( k T ∗ M ) × D b ( k T ∗ M ) −→ D b ( k T ∗ M )( K , K ) K a ◦ K := R p ( p − a K ⊗ p − K ) ≃ R p a ! ( p − a K ⊗ p − a K ) . (2.14)There is a variant of the composition ◦ , constructed in [KS14]: ∗ : D b ( k M ) × D b ( k M ) −→ D b ( k M )( K , K ) K ∗ K := R q ∗ (cid:0) q − ω ⊗ δ !2 ( K ⊠ K ) (cid:1) . (2.15)There is a natural morphism for K ∈ D b ( k M ) and K ∈ D b ( k M ), K ◦ K −→ K ∗ K .Let us state a theorem proven in [KS90, Prop. 4.4.11] refined in [KS14]. Theorem 2.4.
Let F i , G i , H i respectively in D b ( k M ) , D b ( k M ) , D b ( k M ) , i = 1 , .Let U i be an open subset of T ∗ M ij ( i = 1 , , j = i + 1 ) and set U = U a ◦ U . Thereexists a canonical morphism in D b ( k T ∗ M ) , functorial in F (resp. F ): µhom ( F , F ) | U a ◦ µhom ( G , G ) | U −→ µhom ( F ∗ G , F ◦ G ) | U . (2.16) and hence µhom ( F , F ) | U a ◦ µhom ( G , G ) | U −→ µhom ( F ◦ G , F ◦ G ) | U . (2.17)We state the main theorem of this section. Theorem 2.5.
Let F i , G i , H i respectively in D b ( k M ) , D b ( k M ) , D b ( k M ) , i = 1 , then we have: (a) (cid:16) µhom ( F , F ) a ◦ µhom ( G , G ) (cid:17) a ◦ µhom ( H , H ) ∼−→ µhom ( F , F ) a ◦ (cid:16) µhom ( G , G ) a ◦ µhom ( H , H ) (cid:17) The above isomorphism is compatible with the composition ◦ in the sense that thefollowing diagram commutes ( µhom ( F , F ) a ◦ µhom ( G , G )) a ◦ µhom ( H , H ) ∼ / / (cid:15) (cid:15) µhom ( F , F ) a ◦ ( µhom ( G , G ) a ◦ µhom ( H , H )) (cid:15) (cid:15) µhom ( F ◦ G , F ◦ G ) a ◦ µhom ( H , H ) (cid:15) (cid:15) µhom ( F , F ) a ◦ µhom ( G ◦ H , G ◦ H ) (cid:15) (cid:15) µhom (( F ◦ G ) ◦ H , ( F ◦ G ) ◦ H ) ∼ / / µhom ( F ◦ ( G ◦ H ) , F ◦ ( G ◦ H )) Proof. (a) This is a direct application of Lemma 2 . X, Y, Z taken to be respec-tively T ∗ M , T ∗ M , T ∗ M .(b) We shall skip the proof, which is tedious but straightforward. Consider two complex manifolds X and Y of respective dimension d X and d Y . We shallfollow the notations of Section 2.1.For K ∈ D b ( C X × Y ), we recall that we defined the functor Φ K : D b ( C Y ) → D b ( C X ),Φ K ( G ) = Rq ( K ⊗ q − ( G )) , for G ∈ D b ( C Y ). With regards to the notation of Sec-tion 2.1, let us notice that Φ K ( G ) is K ◦ G . We refer also to Section 1.2 for a definitionof Ω X × Y/X .We recall the
Lemma 3.1.
There is a natural morphism Ω X × Y/X ◦ O Y [ d Y ] −→ O X . Proof.
We have Ω X × Y/X ◦ O Y [ d Y ] = R q (Ω X × Y/X ⊗ q − O Y [ d Y ]) −→ R q (Ω X × Y/X [ d Y ]) R −→ O X , where the last arrow is the integration morphism on complex manifolds.The following Lemma will be useful for the proof of Lemma 3.3. Let us first denoteby M i ( i = 1 , , ,
4) four complex manifolds, L i ∈ D b ( C M i,i +1 ), 1 ≤ i ≤
3. We set forshort d i = dim C M i , d ij = d i + d j , Ω ij/i = Ω M ij /M i = Ω (0 ,d j ) M ij . ≤ i ≤ ,K i = µhom ( L i , Ω i,j/i [ d j ]) , j = i + 1 L ij = L i ◦ L j j = i + 1 , L = L ◦ L ◦ L , e K ij = µhom ( L ij , Ω i,j/i [ d j ] ◦ Ω j,k/j [ d k ]) j = i + 1 , k = j + 1 e K = µhom ( L , Ω / [ d ] ◦ Ω / [ d ] ◦ Ω / [ d ]) K ij = µhom ( L ij , Ω i,k/i [ d k ]) j = i + 1 , k = j + 1 ,K = µhom ( L , Ω / [ d ]) . We recall that we have the sequence of natural morphisms:Ω i,j/i ◦ Ω j,k/j = R q i,k ! ( q − i,j Ω i,j/i ⊗ q − j,k Ω j,k/j ) −→ R q i,k ! (Ω i,j,k/i ) −→ Ω i,k/i [ − d j ](3.1) Lemma 3.2.
The following diagram commutes: K ◦ K ◦ K w w ♦♦♦♦♦♦♦♦♦♦♦ ' ' ❖❖❖❖❖❖❖❖❖❖❖ A e K ◦ K / / (cid:15) (cid:15) e K (cid:15) (cid:15) K ◦ e K o o (cid:15) (cid:15) K ◦ K / / B K K ◦ K o o C Proof.
Diagram labelled A commutes by the associativity of the functor µhom (seeTheorem 2.7.3). Let us prove that Diagram B and C commute. Of course, it is enoughto consider Diagram B. To make the notations easier, we assume that M = M = pt.We are reduced to prove the commutativity of the diagram: µhom ( L , Ω ◦ Ω , / [ d ]) ◦ µhom ( L , O ) / / R (cid:15) (cid:15) µhom ( L , Ω ◦ Ω , / [ d ] ◦ O ) R (cid:15) (cid:15) µhom ( L , Ω [ d ]) ◦ µhom ( L , O ) / / µhom ( L , Ω [ d ] ◦ O )For F, F ′ ∈ D b ( k ), G, G ′ ∈ D b ( k ), we saw in Theorem 2.5 (b) that the morphism µhom ( F, F ′ ) ◦ µhom ( G, G ′ ) −→ µhom ( F ◦ G, F ′ ◦ G ′ ) is functorial in F, F ′ , G, G ′ . Thisfact applied to the morphism Ω ◦ Ω , / [ d ] −→ Ω [ d ]gives that the above diagram commutes and so diagram B commutes.15et Z be a complex manifold and let Λ ⊂ T ∗ ( X × Y ) and Λ ′ ⊂ T ∗ ( Y × Z ) be twoconic Lagrangian smooth locally closed complex submanifolds.Let L , L ′ , be perverse sheaves on X × Y , Y × Z , with microsupport SS ( L ) ⊂ Λ, SS ( L ′ ) ⊂ Λ ′ respectively. We set L ′′ := L [ d Y ] ◦ L ′ Assume that p a | Λ : Λ −→ T ∗ Y and p | Λ ′ : Λ ′ −→ T ∗ Y are transversal(3.2)and that the map Λ × T ∗ Y Λ ′ −→ Λ ◦ Λ ′ is an isomorphism.(3.3)Let us set L := µhom ( L, Ω X × Y/X )Note that L ∈ D b ( T ∗ ( X × Y )) is concentrated in degree 0. Indeed, it is proven in[KS90, Th. 10.3.12] that perverse sheaves are the ones which are pure with shift zero atany point of the non singular locus of their microsupport. On the other hand, Theorem9.5.2 of [KS85] together with Definition 9.5.1 of [KS85] show that the latter verify theproperty that, when being applied µhom ( • , Ω X × Y/X ), they are concentrated in degree0. Moreover, L is a ( E X , E Y )-bimodule. Indeed, such actions come from morphism(2.17) and the integration morphism (3.1). We define similarly L ′ and L ′′ .Now consider two open subsets U , V and W of • T ∗ X , • T ∗ Y , • T ∗ Z , respectively.Let K U × V a be the constant sheaf on ( U × V a ) ∩ Λ with stalk H RΓ( U × V a ; L ),extended by 0 elsewhere. K ′ V × W a is the constant sheaf on ( V × W a ) ∩ Λ ′ with stalk H RΓ( V × W a ; L ′ ),extended by 0 elsewhere. K ′′ U × W a is the constant sheaf on ( U × W a ) ∩ Λ ◦ Λ ′ with stalk H RΓ( U × W a ; L ′′ ),extended by 0 elsewhere.Let s, s ′ be sections of Γ( U × V a ; L ) and Γ( V × W a ; L ′ ) respectively. We define theproduct s · s ′ to be the section of Γ( U × W a ; L ′′ ), image of 1 by the following sequenceof morphisms C Λ ◦ Λ ′ ∼ ←− C Λ ◦ C Λ ′ := R p ( p − a C Λ ⊗ p − C Λ ′ ) −→ R p ( p − a K U × V a ⊗ p − K V × W a ) −→ R p ( p − a µhom ( L, Ω X × Y/X ) ⊗ p − µhom ( L ′ , Ω Y × Z/Y )):= µhom ( L, Ω X × Y/X ) ◦ µhom ( L ′ , Ω Y × Z/Y ) −→ L ′′ where the first isomorphism comes from the assumption 3.3. Lemma 3.3.
Assume that conditions 3.2 and 3.3 are satisfied. Let s, s ′ be sections of Γ( U × V a ; L ) and Γ( V × W a ; L ′ ) respectively, and let G ∈ D b ( C Y ) , H ∈ D b ( C Z ) .Then, s defines a morphism α G ( s ) : C Λ ◦ µhom ( G, O Y ) | V −→ µhom ( L [ d Y ] ◦ G, O X ) | U (ii) Considering the morphism α H ( s · s ′ ) : C Λ ◦ Λ ′ ◦ µhom ( H, O Z ) | W −→ µhom ( L [ d Y ] ◦ L ′ [ d Z ] ◦ H, O X ) | U we have the isomorphism α H ( s · s ′ ) ≃ α L ′ [ d Z ] ◦ H ( s ) ◦ Φ C Λ ( α H ( s ′ )) Proof. (i) Given s and two objects G , G ∈ D b ( C Y ), we have a morphism C Λ ◦ µhom ( G , G ) | V −→ µhom ( L ◦ G , Ω X × Y/X ◦ G ) | U corrresponding to the composition of morphisms:R p ( C Λ ⊗ p − a µhom ( G , G ) | V ) −→ R p ( K U × V a ⊗ p − a µhom ( G , G ) | V ) −→ R p ( µhom ( L, Ω X × Y/X ) ⊗ p − a µhom ( G , G ) | V ) −→ µhom ( L ◦ G , Ω X × Y/X ◦ G ) | U (3.4)where the second morphism comes from the natural morphism K U × V a −→ µhom ( L, Ω X × Y/X ).We conclude by choosing, G = G , G = O Y and by using Lemma 3.1: µhom ( L ◦ G , Ω X × Y/X ◦ O Y ) −→ µhom ( L ◦ G , O X [ − d Y ]) ∼−→ µhom ( L [ d Y ] ◦ G , O X )(ii) Let H ∈ D b ( C Z ). We denote by H := µhom ( H, O Z ). It suffices to prove that thefollowing diagram commutes: 17 C Λ ◦ C Λ ′ ) ◦ H ≃ / / ≃ (cid:15) (cid:15) A Φ C Λ ( α H ( s ′ )) ! ! C Λ ◦ ( C Λ ′ ◦ H ) (cid:15) (cid:15) C Λ ◦ C Λ ′ ◦ H α ( s · s ′ ) (cid:25) (cid:25) (cid:15) (cid:15) C Λ ◦ K ′ ◦ H (cid:15) (cid:15) K ◦ K ′ ◦ H (cid:15) (cid:15) C Λ ◦ µhom ( L ′ , Ω Y × Z/Y ) ◦ H (cid:15) (cid:15) C Λ ◦ µhom ( L ′ ◦ H, Ω Y × Z/Y ◦ O Z ) R Z (cid:15) (cid:15) C Λ ◦ µhom ( L ′ [ d Z ] ◦ H, O Y ) (cid:15) (cid:15) α L ′ [ dZ ] ◦ H ( s ) (cid:5) (cid:5) K ◦ µhom ( L ′ , Ω Y × Z/Y ) ◦ H R Z / / (cid:15) (cid:15) B K ◦ µhom ( L ′ [ d Z ] ◦ H, O Y ) (cid:15) (cid:15) µhom ( L, Ω X × Y/X ) ◦ µhom ( L ′ , Ω Y × Z/Y ) ◦ H R Z / / (cid:15) (cid:15) C µhom ( L, Ω X × Y/X ) ◦ µhom ( L ′ [ d Z ] ◦ H, O Y ) (cid:15) (cid:15) µhom ( L ◦ L ′ [ d Z ] ◦ H, Ω X × Y/X ◦ O Y ) R Y (cid:15) (cid:15) µhom ( L ◦ L ′ ◦ H, Ω X × Y/X ◦ Ω Y × Z/Y ◦ O Z ) R Y,Z / / µhom ( L [ d Y ] ◦ L ′ [ d Z ] ◦ H, O X )where we omitted the subscript U × V a and V × W a , H , L ′ [ d Z ] ◦ H for K U × V a , K ′ V × W a , α H , α L ′ [ d Z ] ◦ H , respectively.We know from Theorem 2.4 that the operation ◦ is functorial, so that diagram A B commute. For instance, diagram A decomposes this way: C Λ ◦ C Λ ′ ◦ H ≃ / / (cid:15) (cid:15) C Λ ◦ ( C Λ ′ ◦ H ) (cid:15) (cid:15) C Λ ◦ K ′ ◦ H / / (cid:15) (cid:15) C Λ ◦ µhom ( L ′ , Ω Y × Z/Y ) ◦ H ≃ / / (cid:15) (cid:15) C Λ ◦ ( µhom ( L ′ , Ω Y × Z/Y ) ◦ H ) R Y (cid:15) (cid:15) C Λ ◦ µhom ( L ′ [ d Z ] ◦ H, O Y ) (cid:15) (cid:15) K ◦ K ′ ◦ H / / K ◦ µhom ( L ′ , Ω Y × Z/Y ) ◦ H R Y / / K ◦ µhom ( L ′ [ d Z ] ◦ H, O Y )Besides, diagram C commutes by Lemma 3.2.Finally, the bottom diagonal punctured line correponds to α ( s · s ′ ), since thefollowing diagram commutes C Λ ◦ Λ ′ ≃ / / α ( s · s ′ ) ' ' (cid:15) (cid:15) C Λ ◦ C Λ ′ (cid:15) (cid:15) K ′′ U × W a / / µhom ( L ◦ L ′ [ d Y ] , Ω X × Z/X ) Remark 3.4.
In the following, unless necessary, we will omit the subsript for α . Theorem 3.5.
Let s ∈ Γ( U × V a ; L ) , G ∈ D b ( C Y ) . Then,(i) s defines a morphism α ( s ) : C Λ ◦ µhom ( G, O Y ) | V −→ µhom ( L [ d Y ] ◦ G, O X ) | U . (3.5) (ii) Moreover, if P ∈ Γ( U ; E X ) and Q ∈ Γ( V ; E Y ) satisfy P · s = s · Q , then the diagrambelow commutes C Λ ◦ µhom ( G, O Y ) α ( s ) / / Φ C Λ ( α ( Q )) (cid:15) (cid:15) µhom ( L [ d Y ] ◦ G, O X ) α ( P ) (cid:15) (cid:15) C Λ ◦ µhom ( G, O Y ) α ( s ) / / µhom ( L [ d Y ] ◦ G, O X ) . (3.6) Proof. (i) is already proven in Lemma 3.3.(ii) With regards to the notation of Lemma 3.3, we consider the triplet of manifolds
X, X, Y , Λ = C ∆ X , L := µhom ( C ∆ X [ − n ] , Ω X × X/X ). Then, the assumption 3.2 issatisfied and noticing that Φ C ∆ X ≃ Id X , we conclude by Lemma 3.3 that α ( P ) ◦ α ( s ) ≃ α ( P · s ) ≃ α ( s · Q ) ≃ α ( s ) ◦ Φ C Λ ( α ( Q ))19 .2 Main theorem In this section, we will apply Theorem 3.5 when we are given a homogeneous symplecticisomorphism. Let us recall some useful results.For M a left coherent E X -module generated by a section u ∈ M , we denote by I M the annihilator left ideal of E X given by: I M := { P ∈ E X ; P u = 0 } and by I M the symbol ideal associated to I M : I M := { σ ( P ); P ∈ I M } Definition 3.6 ([Kas03]) . Let M be a coherent E X -module generated by an element u ∈ M . We say that ( M , u ) is a simple E X -module if I M is reduced and I M = { ϕ ∈ O T ∗ X ; ϕ | supp ( M ) = 0 } .Consider two complex manifolds X and Y , open subsets U and V of • T ∗ X and • T ∗ Y ,respectively, and denote by p and p the projections U p ←− U × V a p −→ V . Let Λ be asmooth closed submanifold Lagrangian of U × V a . We will make use of the followingresult from [SKK, Th. 4.3.1], [Kas03, Prop. 8.5]: Theorem 3.7 ([SKK],[Kas03]) . Let ( M , u ) be a simple E X × Y -module defined on U × V a such that supp M = Λ . Assume Λ −→ U is a diffeomorphism. Then, there is anisomorphism of E X -modules: E X | U ∼−→ ( p | U × V a ) ∗ M P P · u Assume that the projections p | Λ and p a | Λ induce isomorphisms. We denote by χ the homogeneous symplectic isomorphism χ := p | Λ ◦ p | − ,Λ ⊂ U × V a ∼ p | Λ w w ♥♥♥♥♥♥♥♥♥♥ ∼ p a | Λ ' ' PPPPPPPPPP • T ∗ X ⊃ U ∼ χ / / V ⊂ • T ∗ Y (3.7) Corollary 3.8.
Let ( M , u ) be a simple E X × Y -module defined on U × V a . Assume supp M = Λ . Then, in the situation of (3 . , we have an anti-isomorphism of algebras χ ∗ E X | U ≃ E Y | V Consider two complex manifolds X and Y of the same dimension n , open subsets U and V of • T ∗ X and • T ∗ Y , respectively, Λ a smooth closed Lagrangian submanifold of U × V a and assume that the projections p | Λ and p a | Λ induce isomorphisms, hence ahomogeneous symplectic isomorphism χ : U ∼−→ V :20 ⊂ U × V a ∼ p w w ♥♥♥♥♥♥♥♥♥♥ ∼ p a ' ' PPPPPPPPPP • T ∗ X ⊃ U ∼ χ / / V ⊂ • T ∗ Y (3.8)We consider a perverse sheaf L on X × Y satisfying( p − ( U ) ∪ p a − ( V )) ∩ SS( L ) = Λ . (3.9)and a section s in Γ( U × V a ; µhom ( L, Ω X × Y/X )).Let G ∈ D b ( C Y ). From Theorem 3.5 (i), the left composition by s defines themorphism α ( s ) in D b ( C U ): C Λ ◦ µhom ( G, O Y ) | V α ( s ) −−→ µhom ( L [ n ] ◦ G, O X ) | U (3.10)The condition (3.9) implies that supp( µhom ( L, Ω X × Y/X ) | p a − ( V ) ) ⊂ Λ. Since, p isan isomorphism from Λ to U and that χ ◦ p | Λ = p a | Λ , we get a morphism in D b ( C U ) χ − µhom ( G, O Y ) | V α ( s ) −−→ µhom (Φ L [ n ] ( G ) , O X ) | U (3.11) Theorem 3.9.
Assume that the section s is non-degenerate on Λ . Then, for G ∈ D b ( C Y ) , we have the following isomorphism in D b ( C U ) χ − µhom ( G, O Y ) | V ∼−→ µhom (Φ L [ n ] ( G ) , O X ) | U (3.12) Moreover, this isomorphism is compatible with the action of E Y and E X on the left andright side of (3.12) respectively.Proof. Let us first prove the following lemma, whose proof is available at the level ofgerms in [KS90, Th. 11.4.9].Let us prove that the morphism (3.11) is an isomorphism. Let L ∗ be the perversesheaf r − R H om ( L, ω X × Y/Y ) where r is the map X × Y −→ Y × X, ( x, y ) ( y, x ). Let s ′ be a section of µhom ( L ∗ , Ω Y × X/Y ), non-degenerate on r (Λ), then we apply the sameprecedent construction to get a natural morphism χ ∗ µhom (Φ L [ n ] ( G ) , O X ) | U −→ µhom (Φ L ∗ [ n ] ◦ Φ L [ n ] G, O Y ) | V ≃ µhom (Φ L ∗ ◦ L [ n ] G, O Y ) | V We know from [KS90, Th. 7.2.1] that C ∆ X ≃ L ∗ ◦ L , so that we get a morphism inD b ( C V ) χ ∗ µhom (Φ L [ n ] ( G ) , O X ) | U α ( s ′ ) −−→ µhom ( G, O Y ) | V (3.13)We must prove that (3.11) and (3.13) are inverse to each other. By Lemma 3.3(ii),we get that the composition of these two morphisms is α ( s ′ · s ), with s ′ · s ∈ E X .21or any left E X -module M , corresponds a right E X -module Ω X ⊗ O X M . Fixing anon-degenerate form t X of Ω X | U (resp. t Y of Ω Y | V ), we apply now Theorem 3.7: s and s ′ are non-degenerate sections so that ( E X × Y | U × V a , t X ⊗ s ) and ( E Y × X | V a × U , s ′ ⊗ t Y )are simple and so isomorphic to ( p − E X | U , ≃ ( p a − E Y | V , X resp. Ω Y beinginvertible O X -module resp. O Y -module, we get as well for the left-right ( E X | U , E Y | V )bi-module, resp. left-right ( E Y | V , E X | U ) bi-module generated by s resp. s ′ , that theyare both isomorphic to p − E X | U ≃ p a − E Y | V .Then, following the proof of [KS90, Th. 11.4.9], s and s ′ , define ring isomorphismsassociating to each P ∈ E X ( U ), P ′ ∈ E X ( U ), some Q ∈ E Y ( V ), Q ′ ∈ E Y ( V ), suchthat P · s = s · Q , s ′ · P ′ = Q ′ · s ′ , respectively. Hence, we get that α ( s ′ ) ◦ α ( s ) is anautomorphism µhom ( G, O Y ) | V , defined by the left action of s ′ · s ∈ E X . Hence, we canchoose s ′ so that α ( s ′ ) ◦ α ( s ) is the identity.We are now in a position to prove Theorem 3.9: we constructed in the proof of thelemma, for each P ∈ p − E X | U , some Q ∈ p − a E Y | V such that P · s = s · Q and we canapply Theorem 3.5 to conclude. We are going to apply the results of the last chapter to the case of projective duality.Recall projective duality for D -modules were performed by D’Agnolo-Schapira [DS96].We will extend their results in a micorlocal setting. In the following, we will quantize the contact transform associated with the Lagrangiansubmanifold • T ∗ S ( P × P ∗ ), where S is the hypersurface of P × P ∗ defined by the incidencerelation h ξ, x i = 0 , ( x, ξ ) ∈ P × P ∗ .We denote by • T ∗ P P , resp. • T ∗ P ∗ P ∗ , the conormal space to P in • T ∗ P , resp. to P ∗ in • T ∗ P ∗ , and we will construct and denote by χ the homogeneous symplectic isomorphismbetween • T ∗ P and • T ∗ P ∗ .For ε ∈ Z / Z , we denote by C P ( ε ) the following sheaves: for ε = 0, we set C P (0) := C P for ε = 1, C P (1) is the sheaf defined by the following exact sequence:0 → C P (1) → q ! C e P tr −→ C P → q is the 2 : 1 map from the universal cover e P of P , to P and tr the integrationmorphism tr : q ! C e P ≃ q ! q ! C P −→ C P .Let an integer p ∈ Z , ε ∈ Z / Z , we define the sheaves of real analytic functions,hyperfunctions on P resp. P ∗ twisted by some power of the tautological line bundle, A P ( ε, p ) := A P ⊗ O P O P ( p ) ⊗ C C P ( ε )22 P ( ε, p ) := B P ⊗ A P A P ( ε, p ) ≃ R H om (D ′ P C P , O P ( p )) ⊗ C P ( ε )We define the sheaves of microfunctions on P resp. P ∗ twisted by some power ofthe tautological bundle, C P ( ε, p ) := H ( µ hom ( D ′ P C P , O P ( p ))) ⊗ C P ( ε )and similarly with P ∗ instead of P . We notice that for n odd, D ′ P C P ≃ C P (0) = C P ,and for n even D ′ P C P ≃ C P (1).For X, Y either the manifold P or P ∗ , for any two integers p, q , we note O X × Y ( p, q )the line bundle on X × Y with homogenity p in the X variable and q in the Y variable.We set Ω X × Y/X ( p, q ) := Ω X × Y/X ⊗ O X × Y O X × Y ( p, q ) E R X ( p, q ) := µhom ( C ∆ X , Ω X × X/X ( p, q ))[ d X ]and we define accordingly E X ( p, q ). Let us notice that E R X ( − p, p ) is a sheaf of rings.Let n be the dimension of P , (of course n = d P ). For an integer k and ε ∈ Z / Z ,we note k ∗ := − n − − kε ∗ := − n − − ε mod (2) We refer to the notations of the sections 1.2. We recall that we denote by V , V , an ( n + 1)-dimensional real and complex vector space, respectively, P , P , the n − dimensional real and complex projective space, respectively, S , S , the real and complex incidence hypersurface in P × P ∗ , P × P ∗ , respectively.When necessary, we will enforce the dimension by noting P n , resp. P ∗ n .Let X, Y be complex manifolds, we recall that we denote by q and q the respectiveprojection of X × Y on each of its factor.For K ∈ D b ( C X × Y ), we recall that we defined the functor:Φ K : D b ( C X ) → D b ( C Y ) F Rq ( K ⊗ q − F )For an integer k and ε ∈ Z / Z , we note k ∗ = − n − − k and ε ∗ = − n − − ε mod(2).We refer to Section 1.2 for the definition of the sheaves of twisted microfunctions C P ( ε, k ) , C P ∗ ( ε ∗ , k ∗ ). 23 .2.2 Geometry of projective duality For a manifold X , we denote by P ∗ X the projectivization of the cotangent bundle of X . The following results are well-known. However, we will give a proof of Prooposition4.1 since it is more straighforward than the one usually found in the litterature. Proposition 4.1.
There is an homogeneous complex symplectic isomorphism • T ∗ P ≃ • T ∗ P ∗ (4.2) and a contact isomorphism P ∗ P ≃ S ≃ P ∗ P ∗ (4.3) Proof.
We have the natural morphism V \ { } ρ −→ P According to 2.3, this morphism, after removing the zero section, induces the followingdiagram T ∗ ( V \ { } ) ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ V \ { } × P T ∗ P t ρ ′ o o / / (cid:15) (cid:15) T ∗ P (cid:15) (cid:15) V \ { } / / P We notice that t ρ ′ is an immersion. Let us denote by H , H ∗ , the incidence hypersurfaces: H = { ( ξ, x ) ∈ V ∗ × ( V \ { } ); h ξ, x i = 0 } H ∗ = { ( x, ξ ) ∈ V × ( V ∗ \ { } ); h x, ξ i = 0 } Noticing that for x ∈ V \ { } , ρ is constant along the fiber above ρ ( x ), we see that t ρ ′ isan immersion into the incidence hypersurface H . Besides, t ρ ′ is a morphism of fiberedspace and so, by a dimensional argument, we conclude that this immersion is also onto.Removing the zero sections, we get the diagram T ∗ ( V \ { } ) ≃ (cid:15) (cid:15) H ? _ o o ≃ (cid:15) (cid:15) V \ { } × P • T ∗ P o o / / ≃ (cid:15) (cid:15) • T ∗ P T ∗ ( V ∗ \ { } ) H ∗ ? _ o o ( V ∗ \ { } ) × P ∗ • T ∗ P ∗ o o / / • T ∗ P ∗ (4.4)where the isomophism between H and H ∗ follows from the following symplectic isomor-phism: • T ∗ ( V \ { } ) ≃ • T ∗ ( V ∗ \ { } )( x, ξ ) ( ξ, − x )24ow, taking the quotient by the action of C ∗ on both sides of the isomorphism between( V \ { } ) × P • T ∗ P and ( V ∗ \ { } ) × P ∗ • T ∗ P ∗ , we get the isomorphism: • T ∗ P ≃ ( V ∗ \ { } ) × P ∗ P ∗ P ∗ ≃ • T ∗ P ∗ This gives (4.2).Besides, passing to the quotient by the action of C ∗ × C ∗ on the two central columnsof diagram (4.4), we get (4.3). Proposition 4.2.
Consider the double fibrations • T ∗ S ( P × P ∗ ) ∼ p y y ssssssssss ∼ p a % % ❑❑❑❑❑❑❑❑❑❑ • T ∗ P ∼ χ / / • T ∗ P ∗ (4.5) Then, p and p a are isomorphisms and χ = p a ◦ p − is a homogeneous symplecticisomorphism. Now, we are going to prove the following
Proposition 4.3.
The diagram 4.5 induces • T ∗ S ( P × P ∗ ) ∩ ( • T ∗ P P × • T ∗ P ∗ P ∗ ) ∼ p u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ ∼ p a ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ • T ∗ P P ∼ χ / / • T ∗ P ∗ P ∗ Let k, k ′ be integers and ε ∈ Z / Z . We follow the notations of the sections 1.2 and 2.We define similarly a twisted version of B ( n, S | P × P ∗ and C ( n, S | P × P ∗ .We set B ( n, S ( k, k ′ ) := q − O P ∗ ( k ′ ) ⊗ q − O P ∗ B S | P × P ∗ ⊗ q − O P q − ( O P ( k ) ⊗ O P Ω P )and the ( E P ( − k, k ) , E P ∗ ( − k ∗ , k ∗ ))-module C ( n, S | P × P ∗ ( k, k ′ ) := E B ( n, S | P × P ∗ ( k, k ′ )We notice that E P ( − k, k ) is nothing but O P ( − k ) D ⊗ π P − D P E P ⊗ π P − D P DO P ( k ). Ac-cording to the diagram 4.5, we denoted by χ the homogeneous symplectic isomorphism χ := p a | • T ∗ S ( P × P ∗ ) ◦ p | − • T ∗ S ( P × P ∗ ) We have: 25 heorem 4.4 ([DS96, p. 469]) . Assume − n − < k < . There exists a section s of µhom ( C S [ − , Ω P × P ∗ / P ∗ ( − k, k ∗ )) , non-degenerate on • T ∗ S ( P × P ∗ ) .Proof. From the exact sequence:0 −→ C ( P × P ∗ ) \ S −→ C P × P ∗ −→ C S −→ P × P ∗ ) \ S ; Ω P × P ∗ / P ∗ ( − k, k ∗ )) −→ RΓ S ( P × P ∗ ; Ω P × P ∗ / P ∗ ( − k, k ∗ ))[1] ≃ RΓ( P × P ∗ ; R H om ( C S ; Ω P × P ∗ / P ∗ ( − k, k ∗ )))[1] ≃ RΓ( T ∗ ( P × P ∗ ); µhom ( C S ; Ω P × P ∗ / P ∗ ( − k, k ∗ )))[1] −→ RΓ( • T ∗ ( P × P ∗ ); µhom ( C S [ − P × P ∗ / P ∗ ( − k, k ∗ )))Let z = ( z , ..., z n ) be a system of homogeneous coordinates on P and ζ = ( ζ , ..., ζ n )the dual coordinates on P ∗ . As explained in [DS96], a non-degenerate section is providedby the Leray section, defined for ( z, ξ ) ∈ ( P × P ∗ ) \ S by s ( z, ζ ) = ω ′ ( z ) h z, ζ i n +1+ k (4.7)where ω ′ ( z ) is the Leray form ω ′ ( z ) = P nk =0 ( − k z k dz ∧ . . . ∧ dz k − ∧ dz k +1 ∧ . . . ∧ dz n ,Leray [Ler59].Let s be a section of H ( µhom ( C S [ − , Ω P × P ∗ / P ∗ ( − k, k ∗ ))), non-degenerate on • T ∗ S ( P × P ∗ ). Theorem 4.5.
Assume − n − < k < . Then, we have an isomorphism in D b ( C • T ∗ P )Φ µ S ( E P ( − k, k ) | • T ∗ P ) ≃ E P ∗ ( − k ∗ , k ∗ ) | • T ∗ P ∗ χ ∗ E P ( − k, k ) | • T ∗ P ≃ E P ∗ ( − k ∗ , k ∗ ) | • T ∗ P ∗ Proof.
Let F , G be line bundles on P , and P ∗ respectively. We know from [SKK] thata global non-degenerate section s ∈ Γ( • T ∗ P × • T ∗ P ∗ ; C ( n, S | P × P ∗ ⊗ p − E P E F ⊗ p − E P ∗ G ⊗− E )induces an isomorphism of E -modulesΦ µ S ( E F | • T ∗ P ) ≃ E G| • T ∗ P ∗ Now, let us set F = O P ( k ), G = O P ∗ ( k ∗ ). Then, 4.4 provides such a non-degeneratesection in Γ( • T ∗ P × • T ∗ P ∗ ; C ( n, S | P × P ∗ ⊗ p − E P E O P ( k ) ⊗ p − E P ∗ O P ∗ ( k ∗ ) ⊗− E ). So that, we havean isomorphism Φ µ S ( E P ( − k, k ) | • T ∗ P ) ≃ E P ∗ ( − k ∗ , k ∗ ) | • T ∗ P ∗ On the other hand s is a non-degenerate section of C ( n, S | P × P ∗ ( − k, k ∗ ), hence we can applyTheorem 3.7. Let us denote by E P × P ∗ ( k, k ∗ ) := E P ( − k, k ) ⊠ π − O P × P ∗ E P ∗ ( − k ∗ , k ∗ )26heorem 3.7 gives the following isomorphisms E P ( − k, k ) | • T ∗ P ≃ p ∗ ( E P × P ∗ ( k, k ∗ ) .s ) | • T ∗ P p ∗ ( E P × P ∗ ( k, k ∗ ) .s ) | • T ∗ P ∗ ≃ E P ∗ ( − k ∗ , k ∗ ) | • T ∗ P ∗ And so χ ∗ E P ( − k, k ) | • T ∗ P ≃ E P ∗ ( − k ∗ , k ∗ ) | • T ∗ P ∗ In the following, we will denote by K the object C S [ n − K ( C P (1)), which is done in [DS96]: Lemma 4.6 ([DS96]) . We have Φ K ( C P (1)) ≃ (cid:26) C P ∗ (1) , for n odd C P ∗ \ P ∗ [1] , for n evenand H j (Φ K ( C P (0))) ≃ C P ∗ , for j = n − C P ∗ \ P ∗ , for j = − and n odd C P ∗ (1) , for j = 0 and n even in any other case We are in a proposition to prove:
Theorem 4.7.
Assume − n − < k < . Recall that any section s ∈ Γ( P × P ∗ ; B ( n, S ( − k, k ∗ )) ,defines a morphism in D b ( C • T P ) χ ∗ C P ( ε, k ) | • T ∗ P P −→ C P ∗ ( ε ∗ , k ∗ ) | • T ∗ P ∗ P ∗ (4.8) Assume s is non-degenerate on • T ∗ S ( P × P ∗ ) . Then (4.8) is an isomorphism. More-over, there exists such a non-degenerate section. Remark 4.8. (i) This is a refinement of a general theorem of [SKK] and is a mi-crolocal version of Theorem 5.17 in [DS96].(ii) The classical Radon transform deals with the case where k = − n , k ∗ = − Proof.
We will deal with the case ε = 1 and n even, the complementary cases beingproven the same way. Let us apply Theorem 2 . U = • T ∗ P , V = • T ∗ P ∗ , Λ = • T ∗ S ( P × P ∗ ).- K is C S [ n − F = C P (1) and F = O P ( k ) K verifies conditions (i),(ii),(iii) of Theorem 2 . SS ( C S ) is nothing but T ∗ S ( P × P ∗ ).(iii) C T ∗ S ( P × P ∗ ) −→ µ hom ( C S , C S ) is an isomorphism on T ∗ S ( P × P ∗ ) (this follows fromthe fact that for a closed submanifold Z of a manifold X , µ Z ( C Z ) ∼−→ C T ∗ Z X , see [KS90,Prop. 4.4.3]).By a fundamental result in [DS96, Th 5.17], we know that for − n − < k < s ∈ Γ( P × P ∗ ; B ( n, S ( − k, k ∗ )), non-degenerate on • T ∗ S ( P × P ∗ ), induces anisomorphism Φ K ( O P ( k )) ≃ O P ∗ ( k ∗ )Formula (4.7) provides an example of such a non-degenerate section. Hence, applyingLemma 4.6, Theorem 2 . χ ∗ µ hom ( C P (1) , O P ( k )) | • T ∗ P ≃ µ hom ( C P ∗ \ P ∗ [1] , O P ∗ ( k ∗ )) | • T ∗ P ∗ We have the exact sequence:0 −→ C P ∗ \ P ∗ −→ C P ∗ −→ C P ∗ −→ F ∈ D b ( C P ∗ ), we have supp ( µhom ( C P ∗ , F ) | • T ∗ P ∗ ) ⊂ ( SS ( C P ∗ ) ∩ • T ∗ P ∗ ) ∩ SS ( F ) = ∅ and hence, µhom ( C P ∗ , F ) | • T ∗ P ∗ ≃ µhom functor to 4 .
9, we get µhom ( C P ∗ \ P ∗ , F ) | • T ∗ P ∗ [ − ≃ µhom ( C P ∗ , F ) | • T ∗ P ∗ Hence, we have proved in particular that χ ∗ µ hom ( C P (1) , O P ( k )) | • T ∗ P P ≃ µ hom ( C P ∗ , O P ∗ ( k ∗ )) | • T ∗ P ∗ P ∗ We follow the notations of Section 1.2 and Section 2.Let us consider the situation (4.5), where we denoted by χ the homogeneous sym-plectic isomorphism between • T ∗ P and • T ∗ P ∗ through • T ∗ S ( P × P ∗ ). We set L := C S [ − L is a perverse sheaf satisfying( p − ( • T ∗ P ) ∪ p a − ( • T ∗ P ∗ )) ∩ SS( L ) = • T ∗ S ( P × P ∗ )(4.10)Recall Theorem 4.4, and let s be a section of µhom ( C S [ − , Ω P × P ∗ / P ∗ ( − k, k ∗ )), non-degenerate on • T ∗ S ( P × P ∗ ). We are in situation to apply Theorem 3.9.28 heorem 4.9. Let G ∈ D b ( C P ∗ ) , k an integer. Assume − n − < k < . Then, wehave an isomorphism in D b ( C • T ∗ P ) : χ − µhom ( G, O P ∗ ( k ∗ )) ∼−→ µhom (Φ C S [ n − ( G ) , O P ( k ))(4.11) This isomorphism is compatible with the action of E P ∗ ( − k ∗ , k ∗ ) and E P ( − k, k ) on theleft and right side of (4.11) respectively.Proof. The isomorphism is directly provided by Theorem 3.9 in the situation where,using the notation inthere, U = • T ∗ P , V = • T ∗ P ∗ and Λ = • T ∗ S ( P × P ∗ ) and where wetwist by homogenous line bundles of P , P ∗ as explained below.Let us adapt (3.12) by taking into account the twist by homogeneous line bundles.We follow the exact same reasoning than sections of 3.1 and 3.2.We have the natural morphismΩ P ∗ × P / P ( − k ∗ , k ) ◦ O P ∗ ( k ∗ ) [ n ] −→ O P ( k ) . Indeed, we haveΩ P ∗ × P / P ( − k ∗ , k ) ◦ O P ∗ ( k ∗ ) [ n ] = R q ( O P ∗ × P ( − k ∗ , k ) ⊗ q − O P ∗ q − Ω P ∗ ⊗ q − O P ∗ ( k ∗ )[ n ]) −→ R q ( O P ∗ × P ( k, ⊗ q − O P ∗ q − Ω P ∗ )[ n ] R −→ O P ( k )Given this morphism and considering L := µhom ( C S [ − , Ω P ∗ × P / P ( − k ∗ , k )), we mimicthe proof of Theorem 3.5 so that for a section s of L on • T ∗ P × • T ∗ P ∗ a and for P ∈ Γ( • T ∗ P ; E P ( − k, k )) and Q ∈ Γ( • T ∗ P ∗ ; E P ∗ ( − k ∗ , k ∗ )) satisfying P · s = s · Q , thediagram below commutes:(4.12) C S ◦ µhom ( G, O P ∗ ( k ∗ )) | • T ∗ P ∗ P ∗ α ( s ) / / Φ CS ( α ( Q )) (cid:15) (cid:15) µhom ( C S [ n − ◦ G, O P ( k )) | • T ∗ P P α ( P ) (cid:15) (cid:15) C S ◦ µhom ( G, O P ∗ ( k ∗ )) | • T ∗ P ∗ P ∗ α ( s ) / / µhom ( C S [ n − ◦ G, O P ( k )) | • T ∗ P P . From there, given a non-degenerate section of L on • T ∗ S ( P × P ∗ ), Theorem 3.9 givesthe compatible action of micro-differential operators on each side of the isomorphism(4.11) χ − µhom ( G, O P ∗ ( k ∗ )) | • T ∗ P ∗ P ∗ ∼−→ µhom (Φ C S [ n − ( G ) , O P ( k )) | • T ∗ P P It remains to exhibit a non-degenerate section so that, for P ∈ Γ( • T ∗ P ; E P ( − k, k )),there is Q ∈ Γ( • T ∗ P ∗ ; E P ∗ ( − k ∗ , k ∗ )) such that P · s = s · Q . Precisely, s is given byProposition 4.4. 29pecializing the above proposition, we get Corollary 4.10.
Let ε ∈ Z / Z . In the situation of Proposition 4.9, we have the iso-morphism, compatible with the respective action of p − E P ( − k, k ) and p a − E P ∗ ( − k ∗ , k ∗ ) χ ∗ C P ( ε, k ) | • T ∗ P P ≃ C P ∗ ( ε ∗ , k ∗ ) | • T ∗ P ∗ P ∗ Proof.
This is an immediate consequence of Proposition 4.9, where we consider thespecial case G = C P ∗ ( ε ∗ ). Indeed, we have, from Lemma 4.6, the isomorphism inD b ( C P ∗ ; • T ∗ P ∗ ) C S [ n − ◦ C P ∗ ( ε ∗ ) ≃ C P ( ε )We can state now Corollary 4.11.
Let k be an integer. Let N be a coherent E P ( − k, k ) -module and F ∈ D b ( P ) . Assume − n − < k < . Then, we have an isomorphism in D b ( C • T ∗ P ) χ ∗ R H om E P ( − k,k ) ( N , µhom ( F, O P ( k ))) ≃ R H om E P ∗ ( − k ∗ ,k ∗ ) (Φ µ S ( N ) , µhom ((Φ C S [ n − F, O P ∗ ( k ∗ ))) Proof.
It suffices to prove this statement for finite free E P ( − k, k )-modules, which inturn can be reduced to the case where N = E P ( − k, k ). By Theorem 4.5, we haveΦ µ S ( E P ( − k, k ) | • T ∗ P ) ≃ E P ∗ ( − k ∗ , k ∗ ) | • T ∗ P ∗ Then, by applying Proposition 4.9, we have χ ∗ µhom ( F, O P ( k )) | • T ∗ P ∗ ≃ R H om E P ∗ ( − k ∗ ,k ∗ ) ( E P ∗ ( − k ∗ , k ∗ ) , µhom ((Φ C S [ n − F, O P ∗ ( k ∗ ))) | • T ∗ P ∗ which proves the corollary. References [Bry86] Jean-Luc Brylinski. “Transformations canoniques, dualit´e projective, th´eoriede Lefschetz, transformations de Fourier et sommes trigonom´etriques”. In:
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