Microlocal theory of sheaves and Tamarkin’s non displaceability theorem

arXiv:1106.1576v2 [math.SG] 15 Feb 2012

Microlocal theory of sheaves and Tamarkin’s non displaceability theorem

St´ephane Guillermou and Pierre Schapira February 15, 2012

Abstract

This paper is an attempt to better understand Tamarkin’s ap- proach of classical non-displaceability theorems of symplectic geome- try, based on the microlocal theory of sheaves, a theory whose main features we recall here. If the main theorems are due to Tamarkin, our proofs may be rather different and in the course of the paper we introduce some new notions and obtain new results which may be of interest.

Introduction

In [12], D. Tamarkin gives a totally new approach for treating classical prob- lems of non-displaceability in symplectic geometry. His approach is based on the microlocal theory of sheaves, introduced and systematically developed in [3, 4, 5]. (Note however that the use of the microlocal theory of sheaves also appeared in a related context in [7, 9, 8].)

The aim of this paper was initially to better understand Tamarkin’s ideas and to give more accessible proofs by making full use of the tools of [5] and of the recent paper [2]. But when working on this subject, we found some new results which may be of interest. In particular, we make here a systematic study of the category of torsion objects.

Let us first briefly recall the main facts of the microlocal theory of sheaves.

Consider a real manifold M of class C^∞ and a commutative unital ring kof finite global dimension. Denote by D^b(kM) the bounded derived category of

sheaves of k-modules on M. In loc. cit. the authors attach to an object F of D^b(kM) its singular support, or microsupport, SS(F), a closed subset of T^∗M, the cotangent bundle to M. The microsupport is conic for the action of R⁺onT^∗M and is involutive (i.e.,co-isotropic). The microsupport allows one to localize the triangulated category D^b(kM), and in particular to define the categoryD^b(kM;U) for an open subsetU ⊂T^∗M. This theory is “conic”, that is, it is invariant by the R⁺-action and is related to the homogeneous symplectic structure rather than the symplectic structure.

In order to get rid of the homogeneity, a classical trick is to add a variable which replaces it. This trick appears for example in the complex case in [10]

where a deformation quantization ring (with an ~-parameter) is constructed on the cotangent bundle T^∗X to a complex manifold X by using the ring of microdifferential operators of [11] on T^∗(X ×C). Coming back to the real setting, denote byta coordinate onR, by (t;τ) the associated coordinates on T^∗R, byT_{{τ >0}}^∗ (M×R) the open subset {τ >0}of T^∗(M×R) and consider the map

ρ:T_{{τ >0}}^∗ (M ×R)→− T^∗M, (x, t;ξ, τ)7→(x;ξ/τ).

Tamarkin’s idea is to work in the localized category D^b(k_M×R;{τ > 0}), the localization of D^b(kM×R) by the triangulated subcategory D^b_{τ≤0}(kM×R) consisting of sheaves with microsupport contained in the set {τ ≤ 0}. He first proves the useful result which asserts that this localized category is equivalent to the left orthogonal to D^b_{τ≤0}(kM×R) and that the convolution by the sheaf k_{t≥0} is a projector on this left orthogonal.

Let us introduce the notation D^b(k^γ_M) :=D^b(kM×R;{τ > 0}) and, for a closed subset A ⊂ T^∗M, let us denote by D^bA(k^γ_M) the full triangulated subcategory of D^b(k^γ_M) consisting of objects with microsupport contained in ρ⁻¹A.

The first result of Tamarkin is a separability theorem. If A and B are two compact subsets of T^∗M, F ∈D^bA(k^γ_M), G∈D^bB(k^γ_M), and ifA∩B =∅, then Hom_Db(k^γ_M)(F, G)≃0.

The second result of Tamarkin is a Hamiltonian isotopy invariance the- orem, up to torsion, that is, after killing what he calls the torsion objects.

An object F ∈D^b(k^γ_M) is torsion if there exists c≥ 0 such that the natural map F −→Tc∗(F) is zero, Tc∗(F) denoting the image of F by the translation t 7→t+c in the t-variable. Let I be an open interval of R containing [0,1]

and let Φ ={ϕs}s∈I be a Hamiltonian isotopy (with ϕ0 = id) such that there

exists a compact set C ⊂T^∗M satisfying ϕs|T^∗M\C = idT^∗M\C for all s∈ I.

Tamarkin constructs a functor Ψ : D^bA(k^γ_M)−→D^b_ϕ1(A)(k^γ_M) such that Ψ(F) is isomorphic to F modulo torsion, for any F ∈D^bA(k^γ_M).

From these two results he easily deduces that if A, B ⊂ T^∗M are com- pact sets and if there exist F ∈ D^bA(k^γ_M), G ∈ D^bB(k^γ_M) such that the map RHom_Db(k^γ_M)(F, G)−→ RHom_Db(k^γ_M)(F, T_c(G)) is not zero for all c≥ 0, then the setsAand B are mutually non displaceable, that is, for any Hamiltonian isotopy Φ as above and any s∈I, A∩ϕs(B)6=∅.

Let us describe the contents of this paper.

In Section 1 we recall some constructions and results of [5] on the mi- crolocal theory of sheaves.

In Section 2 we recall the main theorem of [2] which allows one to quantize homogeneous Hamiltonian isotopies and we also give some geometrical tools relying homogeneous and non homogeneous symplectic geometry.

In Section 3 we study convolution of sheaves on a trivial vector bundle E = M ×V over M as well as the category D^b(kE;Uγ), the localization of the category D^b(kE) onUγ =E×V ×Int(γ₀^◦) where Int(γ₀^◦) is the interior of the polar cone to a closed convex proper coneγ₀ inV. We prove in particular a separability theorem in this category.

In Section 4 we introduce the Tamarkin category D^b(k^γ_M), that is, the category D^b(kE;Uγ) for E =M×R and γ0={t ≥0}.

In Section 5 we make a systematic study of the category N_tor of torsion objects, proving that this category is triangulated and also proving that, under some hypothesis on the microsupport, an object is torsion if and only if its restriction to one point is torsion (Theorem 5.12).

Finally, in Section 6 we give a proof of the Hamiltonian isotopy invariance theorem of Tamarkin. The existence of the functor Ψ mentioned above is now an easy consequence on the results of [2], and one checks that this functor induces a functor isomorphic to the identity functor modulo torsion.

As already mentioned, Tamarkin’s non displaceability theorem is an easy corollary of the preceding results.

Note that, for the purposes we have in mind, we do not need to consider the unbounded derived category D(kM), as did Tamarkin, but only its full triangulated category D^lb(kM) consisting of locally bounded objects. Also note that our notations, as well as our proofs, may seriously differ from Tamarkin’s ones.

In a next future, motivated by the papers of Fukaya-Seidel-Smith [1] and Nadler [8], we plan to use the tools developed here to study sheaves associated with smooth Lagrangian manifolds.

Acknowledgment We have been very much stimulated by the interest of Claude Viterbo for the applications of sheaf theory to symplectic topology and it is a pleasure to thank him here. We also thank Masaki Kashiwara for many enlightning discussions.

1 Microlocal theory of sheaves

In this section, we recall some definitions and results from [5], following its notations with the exception of slight modifications. We consider a real manifold M of class C^∞.

Some geometrical notions ([5, § 4.2, § 6.2])

For a locally closed subset A ofM, one denotes by Int(A) its interior and by A its closure. One denotes by ∆M or simply ∆ the diagonal of M×M.

One denotes by τ: T M −→M and π: T^∗M −→M the tangent and cotan- gent bundles to M. If L⊂M is a (smooth) submanifold, we denote byTLM its normal bundle and T_L^∗M its conormal bundle. They are defined by the exact sequences

0−→T L−→L×_M T M −→TLM −→0, 0−→T_L^∗M −→L×_M T^∗M −→T^∗L−→0.

One identifiesM toT_M^∗M, the zero-section ofT^∗M. One sets ˙T^∗M:=T^∗M\ T_M^∗ M and one denotes by ˙πM: ˙T^∗M −→M the projection.

Letf: M −→N be a morphism of real manifolds. Tof are associated the tangent morphisms

T M

τ

f^′

//M ×_N T N

τ

fτ

//T N

τ

M M ^{f //}N.

(1)

By duality, we deduce the diagram:

T^∗M

π

M ×_N T^∗N

π

fd

oo ^fπ //T^∗N

π

M M ^{f //}N.

(2)

One sets

T_M^∗ N:= Kerfd=f_d⁻¹(T_M^∗ M).

Note that, denoting by Γf the graph off inM×N, the projectionT^∗(M× N)−→M ×T^∗N identifies T_Γ^∗_f(M ×N) and M×_N T^∗N.

For two subsets S1, S2 ⊂ M, their Whitney’s normal cone, denoted C(S1, S2), is the closed cone of T M defined as follows. Let (x) be a lo- cal coordinate system and let (x;v) denote the associated coordinate system on T M. Then





(x0;v0) ∈ C(S1, S2) ⊂ T M if and only if there exists a se- quence {(x_n, yn, cn)}_n ⊂ S1 ×S2 ×R⁺ such that xn

−n

→ x0, yn n

−

→x0 and cn(xn−yn)−→ⁿ v0.

For a subset SofM and a smooth closed submanifoldLofM, the Whitney’s normal cone of S along L, denoted CL(S), is the image in TLM of C(L, S).

If L={p}, we write Cp(S) instead of C{p}(S).

Now consider the homogeneous symplectic manifoldT^∗M: it is endowed with the Liouville 1-form given in a local homogeneous symplectic coordinate system (x;ξ) on T^∗M by

αM =hξ, dxi.

The antipodal map a_M is defined by:

aM: T^∗M −→T^∗M, (x;ξ)7→(x;−ξ).

(3)

If A is a subset ofT^∗M, we denote by A^a instead of aM(A) its image by the antipodal map.

We shall use the Hamiltonian isomorphism H: T^∗(T^∗M) −→∼ T(T^∗M) given in a local symplectic coordinate system (x;ξ) by

H(hλ, dxi+hµ, dξi) =−hλ, ∂_ξi+hµ, ∂_xi.

Definition 1.1. (see [5, Def. 6.5.1]) A subset S of T^∗M is co-isotropic (one also says involutive) atp∈T^∗M if for anyθ ∈T_p^∗T^∗M such that the Whitney normal coneCp(S, S) is contained in the hyperplane{v ∈T T^∗M;hv, θi= 0}, one has −H(θ)∈Cp(S). A set S is co-isotropic if it is so at each p∈S.

When S is smooth, one recovers the usual notion.

Microsupport

We consider a commutative unital ring k of finite global dimension (e.g.

k = Z). We denote by D(k_M) (resp. D^b(k_M)) the derived category (resp.

bounded derived category) of sheaves of k-modules on M.

Recall the definition of the microsupport (or singular support) SS(F) of a sheaf F.

Definition 1.2. (see [5, Def. 5.1.2]) Let F ∈D^b(kM) and let p∈T^∗M. One says that p /∈ SS(F) if there exists an open neighborhood U of p such that for any x₀ ∈M and any realC¹-function ϕ onM defined in a neighborhood of x0 satisfying dϕ(x0)∈U and ϕ(x0) = 0, one has (RΓ{x;ϕ(x)≥0}(F))x0 ≃0.

In other words, p /∈ SS(F) if the sheaf F has no cohomology supported by “half-spaces” whose conormals are contained in a neighborhood of p.

• By its construction, the microsupport is closed and is R⁺-conic, that is, invariant by the action of R⁺ onT^∗M.

• SS(F)∩T_M^∗M =πM(SS(F)) = Supp(F).

• The microsupport satisfies the triangular inequality: if F1 −→ F2 −→ F₃ −→⁺¹ is a distinguished triangle in D^b(k_M), then SS(F_i) ⊂ SS(F_j)∪ SS(Fk) for alli, j, k ∈ {1,2,3} with j 6=k.

Theorem 1.3. (see [5, Th. 6.5.4]) Let F ∈D^b(kM). Then its microsupport SS(F) is co-isotropic.

In the sequel, for a locally closed subsetZinM, we denote bykZ the constant sheaf with stalk k onZ, extended by 0 onM \Z.

Example 1.4. (i) If F is a non-zero local system on a connected manifold M, then SS(F) = T_M^∗ M, the zero-section.

(ii) If N is a smooth closed submanifold of M and F = kN, then SS(F) = T_N^∗M, the conormal bundle to N in M.

(iii) Let ϕ be C¹-function with dϕ(x) 6= 0 when ϕ(x) = 0. Let U = {x ∈ M;ϕ(x)>0}and let Z ={x∈M;ϕ(x)≥0}. Then

SS(kU) =U ×_M T_M^∗M ∪ {(x;λdϕ(x));ϕ(x) = 0, λ≤0}, SS(kZ) =Z×_M T_M^∗ M∪ {(x;λdϕ(x));ϕ(x) = 0, λ≥0}.

(iv) Let (X,O_X) be a complex manifold and let M be a coherent module over the ring D_X of holomorphic differential operators. (Hence, M rep- resents a system of linear partial differential equations on X.) Denote by F = RHomD_X(M,O_X) the complex of holomorphic solutions ofM. Then SS(F) = char(M), the characteristic variety ofM.

Functorial operations (proper and non-characteristic cases)

Let M and N be two real manifolds. We denote by qi (i = 1,2) the i-th projection defined on M×N and by pi (i= 1,2) the i-th projection defined on T^∗(M ×N)≃T^∗M×T^∗N.

Definition 1.5. Let f: M −→ N be a morphism of manifolds and let Λ ⊂ T^∗N be a closed R⁺-conic subset. One says that f is non-characteristic for Λ (or else, Λ is non-characteristic for f, or f and Λ are transversal) if

f_π⁻¹(Λ)∩T_M^∗ N ⊂M ×N T_N^∗N.

A morphismf:M −→N is non-characteristic for a closedR⁺-conic subset Λ of T^∗N if and only if f_d: M×_N T^∗N −→ T^∗M is proper on f_π⁻¹(Λ) and in this case fdf_π⁻¹(Λ) is closed and R⁺-conic in T^∗M.

We denote by ωM the dualizing complex on M. Recall that ωM is iso- morphic to the orientation sheaf shifted by the dimension. We also use the notation ωM/N for the relative dualizing complex ωM ⊗f⁻¹ω_N^⊗−1. We have the duality functors

DM(^•) = RHom(^•, ωM), (4)

D^′_M(^•) = RHom(^•,kM).

(5)

Theorem 1.6. (See [5,§5.4].) Let f: M −→N be a morphism of manifolds, let F ∈D^b(k_M) and let G∈D^b(k_N).

(i) One has

SS(F ⊠^L G)⊂SS(F)×SS(G),

SS(RHom(q₁⁻¹F, q₂⁻¹G))⊂SS(F)^a×SS(G).

(ii) Assume that f is proper on Supp(F). Then SS(Rf!F)⊂fπf_d⁻¹SS(F). (iii) Assume that f is non-characteristic with respect to SS(G). Then the

natural morphismf⁻¹G⊗ωM/N −→f^!(G)is an isomorphism. Moreover SS(f⁻¹G)∪SS(f^!G)⊂fdf_π⁻¹SS(G).

(iv) Assume that f is smooth (that is, submersive). Then SS(F) ⊂ M ×N

T^∗N if and only if, for anyj ∈ Z, the sheavesH^j(F)are locally constant on the fibers off.

For the notion of a cohomologically constructible sheaf we refer to [5,

§ 3.4].

Corollary 1.7. Let F1, F2 ∈D^b(kM).

(i) Assume that SS(F₁)∩SS(F₂)^a ⊂T_M^∗ M. Then

SS(F₁⊗^L F₂)⊂SS(F₁) + SS(F₂).

(ii) Assume that SS(F₁)∩SS(F₂)⊂T_M^∗ M. Then

SS(RHom(F1, F2))⊂SS(F1)^a+ SS(F2).

Moreover, assuming thatF1 is cohomologically constructible, the natural morphism D^′F1

⊗L F2 −→RHom(F1, F2) is an isomorphism.

The next result follows immediately from Theorem 1.6 (ii). It is a partic- ular case of the microlocal Morse lemma (see [5, Cor. 5.4.19]), the classical theory corresponding to the constant sheaf F =kM.

Corollary 1.8. Let F ∈ D^b(k_M), let ϕ: M −→ R be a function of class C¹ and assume that ϕ is proper on supp(F). Let a < b in R and assume that dϕ(x)∈/SS(F) for a≤ϕ(x)< b. Then the natural morphism

RΓ(ϕ⁻¹(]− ∞, b[);F)−→ RΓ(ϕ⁻¹(]− ∞, a[);F) is an isomorphism.

Corollary 1.9. Let I be a contractible manifold and let p: M ×I −→ M be the projection. If F ∈ D^b(kM×I) satisfies SS(F) ⊂ T^∗M ×T_I^∗I, then F ≃p⁻¹Rp∗F.

Proof. It follows from Theorem 1.6 (iv) that the restriction F|{x}×I is locally constant for any x∈M. Then the result follows from [5, Prop. 2.7.8].

Corollary 1.10. Let I be an open interval of R and let q: M ×I −→ I be the projection. Let F ∈ D^b(kM×I) such that SS(F)∩(T_M^∗ M × T^∗I) ⊂ T_M×I^∗ (M × I) and q is proper on Supp(F). Then we have isomorphisms RΓ(M;F_s)≃ RΓ(M;F_t) for any s, t∈I.

Proof. It follows from Theorem 1.6 that SS(Rq∗(F)) ⊂ T_I^∗I. Hence, there exists V ∈D^b(k) and an isomorphism Rq∗(F)≃VI. (Recall thatVI =a⁻¹_I V, where a_I −→ {pt} is the projection and V is identified to a sheaf on {pt}.) Since we have RΓ(M;Fs)≃(Rq∗(F))s the result follows.

Kernels ([5, § 3.6])

Notation 1.11. Let Mi (i = 1,2,3) be manifolds. For short, we write Mij:=Mi×M_j (1≤i, j ≤3) andM123 =M1×M2×M₃. We denote byqi the projection M_ij −→M_i or the projectionM₁₂₃ −→M_i and by q_ij the projection M123 −→Mij. Similarly, we denote bypi the projectionT^∗Mij −→T^∗Mi or the projection T^∗M123 −→T^∗Mi and bypij the projection T^∗M123−→T^∗Mij. We also need to introduce the mapp₁₂^a, the composition ofp₁₂ and the antipodal map on T^∗M2.

LetA⊂T^∗M12 and B ⊂T^∗M23. We set A×T^∗M2a B =p⁻¹₁₂(A)∩p⁻¹₂^a₃(B) A◦^aB =p13(A×T^∗M2a B)

={(x₁, x3;ξ1, ξ3)∈T^∗M13; there exists (x2;ξ2)∈T^∗M2, (x₁, x₂;ξ₁, ξ₂)∈A,(x₂, x₃;−ξ₂, ξ₃)∈B}.

(6)

We consider the operation of composition of kernels:

◦: D^b(kM12)×D^b(kM23) −→ D^b(kM13)

(K₁, K₂) 7→ K₁◦K₂:= Rq_13!(q₁₂⁻¹K₁⊗^L q₂₃⁻¹K₂).

(7)

Let Ai = SS(Ki)⊂T^∗Mi,i+1 and assume that









(i) q13 is proper on q₁₂⁻¹supp(K1)∩q⁻¹₂₃ supp(K2), (ii) p⁻¹₁₂A1∩p⁻¹₂^a₃A2∩(T_M^∗₁M1×T^∗M2×T_M^∗₃M3)

⊂T_M^∗_1×M2×M3(M1×M2×M3).

(8)

It follows from Theorem 1.6 that under the assumption (8) we have:

SS(K1◦K2)⊂A1

◦aA2. (9)

Characteristic inverse images

Theorem 1.6 treats the easy cases of external tensor product or external Hom , non-characteristic inverse images or proper direct image. In order to treat more general cases we introduce some additional geometrical notions.

Let Λ be a smooth Lagrangian submanifold of T^∗M. The Hamiltonian isomorphism defines an isomorphism

T^∗Λ≃TΛT^∗M.

Let j: L ֒→ M be the embedding of a smooth submanifold L of M. The Liouville form defines an embedding

T^∗L ֒→T^∗T_L^∗M ≃TT_L^∗MT^∗M.

Now consider a morphism of manifolds f: M −→N and let us identify M to the graph of f inM ×N. For a subset B ⊂T^∗N one sets:

f^♯(B) =T^∗M ∩CT_M^∗(M×N)(T_M^∗ M ×B).

(10)

In local symplectic coordinate systems (x;ξ) on M and (y;η) on N one has





(x0;ξ0) ∈ f^♯(B) if and only if there exist sequences {x_n}_n⊂M and {(y_n;ηn)}_n⊂B such that

xn −→x0, ^tf^′(xn)·ηn

−n

→ξ0 and |y_n−f(xn)| · |η_n|−→ⁿ 0.

(11)

For two closed R⁺-conic subsets A and B of T^∗M one sets A+b B =T^∗M ∩C(A, B^a).

(12)

Here, C(A, B^a) is considered as a subset of T^∗T^∗M via the Hamiltonian isomorphism and T^∗M is embedded into T^∗T^∗M via the Liouville form αM. In a local coordinate system, one has









(z0;ζ0) ∈ A +b B if and only if there exist se- quences{(x_n;ξn)}_n inA and {(y_n;ηn)}_n inB such that xn

−n

→ z0, yn

−n

→ z0, ξn + ηn

−n

→ ζ0 and

|xn−yn| · |ξn|−→ⁿ 0.

(13)

Theorem 1.12. (See [5, Cor. 6.4.4, 6.4.5].) Let F1, F2 ∈ D^b(kM) and let G∈D^b(kN). Then

SS(F1

⊗L F2)⊂SS(F1)+ SS(Fb 2),

SS(RHom(F1, F2))⊂SS(F2)+ SS(Fb 1)^a, SS(f⁻¹G)∪SS(f^!G)⊂f^♯(SS(G)).

Non proper direct images

We shall also need a direct image theorem in a non proper case.

Consider a constant linear map u of trivial vector bundles over M, that is, we assume that Ei =M ×Vi (i = 1,2) and u: V1 −→ V2 is a linear map.

The map u defines the maps described by the diagram T^∗M×V1×V₂^∗

ud

uu❦❦❦❦❦❦❦❦❦❦❦❦❦❦ _uπ

))❙

❙❙

❙❙

❙❙

❙❙

❙❙

❙❙

❙❙

T^∗M×V1×V₁^∗

vπ

))

❙❙

❙❙

❙❙

❙❙

❙❙

❙❙

❙❙

T^∗M×V2 ×V₂^∗.

vd

uu❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦

T^∗M×V2×V₁^∗ Note that for a subset A of T^∗E1 we have

uπ(u⁻¹_d (A)) =v⁻¹_d (vπ(A)).

(14)

Notation 1.13. Letu:E1 −→E2 be a constant linear map of trivial vector bundles over M and let A⊂T^∗E1 be a closed subset. We set

u♯(A) = v_d⁻¹(vπ(A)).

(15)

In Lemmas 1.14 and 1.15 below we use the notationsL

nGn and Q

nGn

for a family {G_n}_n∈N in D^b(kM). We define it as follows. Let p: M ×N −→ M be the projection. Then we have a unique G ∈ D^b(kM×N) such that G|_M×{n} ≃Gn, for all n, and we set L

nGn:= Rp!Gand Q

nGn:= Rp∗G.

Lemma 1.14. Let M be a manifolds and let {U_n}_n∈N be an increasing se- quence of open subsets of M such that M = S

nU_n. Then, for any F ∈ D^b(kM), we have the distinguished triangles

M

n

F_Un −−−→^id−s1 M

n

F_Un −→F −→, F⁺¹ −→Y

n

RΓ_Un(F)−−−→^id−s2 Y

n

RΓ_Un(F)−→,⁺¹ wheres1 is the sum of the natural morphismsFUn −→FUn+1 ands2the product of the natural morphisms RΓ_Un+1(F) −→ RΓ_Un(F) for n ≥ 0 and the zero morphism for n =−1.

Proof. These triangles arise from similar exact sequences of sheaves when F is a flabby sheaf. The exactness can be checked easily on the stalks in the first case and on sections over any open subset in the second case.

Lemma 1.15. Let f: M −→N be a morphism of manifolds and let{U_n}_n∈N

be an increasing sequence of open subsets of M such that M =S

nUn. Then, for any F ∈D^b(kM), we have

SS(Rf!F)⊂[

n

SS(Rf!(FUn)), SS(Rf∗F)⊂[

n

SS(Rf∗RΓUn(F)).

Proof. We can check, similarly as in [5, Exe. V.7], that for any family {Gn}n∈N in D^b(kN) we have SS(L

nGn)∪SS(Q

nGn) ⊂ S

nSS(Gn). Then the result follows from Lemma 1.14 and the fact that Rf! commutes with ⊕ and Rf_∗ with Q

.

The following result is due to Tamarkin [12, Lem. 3.3] but our proof is completely different.

Theorem 1.16. Let u: E1 −→ E2 be a constant linear map of trivial vector bundles over M and let F ∈ D^b(kE1). Then SS(Ru!F) ⊂ u♯(SS(F)). The same estimate holds with Ru!F replaced with Ru∗F.

Proof. (i) By decomposing uby its graph, one is reduced to prove the result for an immersion and for a projection. Since the case of an immersion is obvious, we restrict ourselves to the case where E =M ×V and u:E −→M is the projection. Moreover the result is local onM and we may assume that M is an open subset in a vector space W.

(ii) We consider (x0;ξ0) ∈ T^∗M ≃ M ×W^∗ such that (x0;ξ0) 6∈ u♯(SS(F)).

We will prove that (x0;ξ0)6∈SS(Ru!F)∪SS(Ru∗F). Ifξ0 = 0, thenF|U×V ≃ 0 for some neighborhood U of x0 and the result follows easily. Hence we assume that ξ0 6= 0. Up to shrinking M we may find an open cone C ⊂ W^∗×V^∗ such that (ξ0,0)∈C and SS(F)∩((M ×V)×C) =∅.

(iii) We choose an open convex cone γ ⊂W ×V such that γ∩({0} ×V) = {(0,0)}and γ^◦ ⊂ C. We also choose two sequences of points {z_n}_n∈N, resp.

{z_n^′}n∈N, of W ×V such that W ×V is the increasing union of the cones γn =zn−γ, resp. γ_n^′ =z_n^′ +γ. By Lemma 1.15 it is enough to show

(SS(Ru_∗RΓ_γnF)∪SS(Ru_!(F_γn^′)))∩(M ×(C∩(W^∗× {0})) =∅.

(iv) By Lemma 3.16 below SS(kγn)⊂(W×V)×(−C). Using D_M^′ (kγn)≃kγn

we deduce SS(kγn)⊂ (W ×V)×C. Similarly SS(kγn^′)⊂(W ×V)×(−C).

Since SS(F)∩((M ×V)×C) = ∅, Corollary 1.7 gives

(SS(RΓγnF)∪SS((Fγ_n^′)))∩((M ×V)×C) = ∅.

Since γ ∩({0} ×V) = {(0,0)} the mapu:M ×V −→M is proper on all γ_n and γ^′_n and the result follows from Theorem 1.6 (ii).

For a trivial vector bundleE =M ×V we denote by b

πE: T^∗E −→T^∗M ×V^∗, (16)

or bπ if there is no risk of confusion, the natural projection. We say that a subset of T^∗M ×V^∗ is a cone if it is stable by the multiplicative action of R⁺ given by

λ·(x;ξ, v) = (x;λξ, λv).

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We will be mainly concerned with the case where F ∈ D^b(kE) has a micro- support bounded by πb_E⁻¹(A) for some closed cone A⊂T^∗M ×V^∗.

Letu: E1 =M ×V1 −→E2 =M ×V2 be a constant linear map of trivial vector bundles over M and denote by

e

u_d: T^∗M ×V₂^∗ −→T^∗M ×V₁^∗ (18)

the map associated with u.

Corollary 1.17. Let u: E1 −→E2 be a constant linear map of trivial vector bundles over M and let F ∈ D^b(kE1). Assume that SS(F) ⊂ πb_E⁻¹₁(A1) for a closed cone A1 ⊂ T^∗M ×V₁^∗. Then SS(Ru!F) ⊂ bπ_E⁻¹₂ue⁻¹_d (A1). The same estimate holds with Ru!F replaced with Ru∗F.

Proof. We have vπ(bπ⁻¹_E1(A1)) = A1×V2 and this set is closed. We thus have u♯(bπ_E⁻¹₁(A1)) = v⁻¹_d (vπ(πb⁻¹_E1(A1))) = uπ(u⁻¹_d (A1×V1))

= eu⁻¹_d (A1)×V2 =bπ⁻¹_E2ue⁻¹_d (A1).

Localization

LetT be a triangulated category,N a null system, that is, a full triangulated subcategory with the property that if one has an isomorphism F ≃G inT with F ∈ N , then G ∈ N . The localization T /N is a well defined triangulated category (we skip the problem of universes). Its objects are those of T and a morphism u: F₁ −→ F₂ in T becomes an isomorphism in T /N if, after embedding this morphism in a distinguished triangle F1 −→ F2 −→F3

−→, one has+1 F3 ∈N .

Recall that the left orthogonal N ^⊥,l of N is the full triangulated sub- category of T defined by:

N ^⊥,l ={F ∈T; HomT(F, G)≃0 for all G∈N }.

By classical results (see e.g., [6, Exe. 10.15]), if the embedding N ^⊥,l ֒→ T admits a left adjoint, or equivalently, if for any F ∈ T, there exists a distinguished triangle F^′ −→F −→F^′′ −→⁺¹ withF^′ ∈N ^⊥,l andF^′′∈N , then there is an equivalence N ^⊥,l ≃T /N .

Of course, there are similar results with the right orthogonalN ^⊥,r.

Now letU be a subset ofT^∗M and setZ =T^∗M\U. The full subcategory D^bZ(kM) of D^b(kM) consisting of sheaves F such that SS(F) ⊂ Z is a null system. One sets

D^b(kM;U) :=D^b(kM)/D^bZ(kM),

the localization of D^b(kM) by D^bZ(kM). Hence, the objects ofD^b(kM;U) are those of D^b(kM) but a morphism u: F1 −→ F2 in D^b(kM) becomes an iso- morphism inD^b(kM;U) if, after embedding this morphism in a distinguished triangle F1 −→F2 −→F3

−→, one has SS(F+1 3)∩U =∅.

For a closed subsetA ofU, D^bA(kM;U) denotes the full triangulated sub- category of D^b(kM;U) consisting of objects whose microsupports have an intersection with U contained in A.

Quantized symplectic isomorphisms ([5, §7.2])

Consider two manifolds M and N, two conic open subsets U ⊂ T^∗M and V ⊂T^∗N and a homogeneous symplectic isomorphism χ:

T^∗N ⊃V −→∼

χ U ⊂T^∗M.

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Denote by V^a the image of V by the antipodal map aN on T^∗N and by Λ the image of the graph of ϕ by idU×aN. Hence Λ is a conic Lagrangian submanifold of U ×V^a. A quantized contact transformation (a QCT, for short) above χ is a kernel K ∈D^b(kM×N) such that SS(K)∩(U ×V^a)⊂Λ and satisfying some technical properties that we do not recall here, so that the kernel K induces an equivalence of categories

K◦ ^•: D^b(kN;V)−→∼ D^b(kM;U).

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Given χ and q ∈ V, p =χ(q)∈ U, there exists such a QCT after replacing U and V by sufficiently small neighborhoods ofp and q.

Simple sheaves ([5, §7.5])

Let Λ⊂T˙^∗M be a locally closed conic Lagrangian submanifold and letp∈Λ.

Simple sheaves along Λ at p are defined in [5, Def. 7.5.4].

When Λ is the conormal bundle to a submanifold N ⊂M, that is, when the projectionπM|Λ: Λ−→M has constant rank, then an objectF ∈D^b(kM) is simple along Λ at p if F ≃kN[d] in D^b(kM;p) for some shift d∈Z.

If SS(F) is contained in Λ on a neighborhood of Λ, Λ is connected andF is simple at some point of Λ, then F is simple at every point of Λ.

The functor µhom ([5, §4.4, §7.2])

The functor of microlocalization along a submanifold has been introduced by Mikio Sato in the 70’s and has been at the origin of what is now called

“microlocal analysis”. A variant of this functor, the bifunctor µhom: D^b(k_M)^op×D^b(k_M)−→D^b(k_T^∗_M) (21)

has been constructed in [5]. Let us only recall the properties of this functor that we shall use. ForF, G∈D^b(kM), withF cohomologically constructible, we have

RπM∗µhom(F, G) ≃ RHom(F, G), RπM!µhom(F, G) ≃ D^′_M(F)⊗^L G and we deduce the distinguished triangle

D^′_M(F)⊗^L G−→RHom(F, G)−→R ˙πM∗(µhom(F, G)|T˙^∗M)−→⁺¹ . (22)

Let Λ ⊂T˙^∗M be a locally closed smooth conic Lagrangian submanifold and let F ∈D^b(kM) be simple along Λ. Then

µhom(F, F)|_Λ≃k_Λ. (23)

2 Quantization of Hamiltonian isotopies

In this section, we recall the main theorem of [2].

We first recall some notions of symplectic geometry. LetXbe a symplec- tic manifold with symplectic form ω. We denote by X^a the same manifold endowed with the symplectic form −ω. The symplectic structure induces the Hamiltonian isomorphismh: TX−→∼ T^∗Xbyh(v) =ι_v(ω), where ι_v denotes the contraction with v (in case Xis a cotangent bundle we have h=−H⁻¹, where H is used in Definition 1.1). To a vector field v onX we associate in this way a 1-formh(v) onX. For aC^∞-functionf: X−→R, the Hamiltonian vector field of f is by definition Hf :=−h⁻¹(df).

A vector field v is called symplectic if its flow preserves ω. This is equiv- alent to L_v(ω) = 0 where L_v denotes the Lie derivative of v. By Cartan’s formula (L_v = d ιv +ιvd) this is again equivalent to d(h(v)) = 0 (recall that dω = 0). The vector field v is called Hamiltonian if h(v) is exact, or equivalently v =Hf for some function f on X.

LetI be an open interval ofRcontaining the origin and let Φ : X×I −→X be a map such thatϕs:=Φ(·, s) : X−→Xis a symplectic isomorphism for each s ∈ I and is the identity for s = 0. The map Φ induces a time dependent vector field on X

v_Φ:=∂Φ

∂s :X×I −→ TX. (24)

The “time dependent” 1-formβ =h(v_Φ) : X×I −→T^∗Xsatisfiesd(β_s) = 0 for any s∈I. The map Φ is called a Hamiltonian isotopy if vΦ,s is Hamiltonian, that is, if βs is exact, for any s. In this case we can write βs = −d(f_s) for some C^∞-functionf: X×I −→R. Hence we have

∂Φ

∂s =Hfs.

The fact that the isotopy Φ is Hamiltonian can be interpreted as a geo- metric property of its graph as follows. For a given s ∈ I we let Λs be the graph of ϕ⁻¹_s and we let Λ^′ be the family of Λ_s’s:

Λs ={(ϕ_s(v), v) ;v ∈X^a} ⊂X×X^a,

Λ^′ ={(ϕs(v), v, s) ;v ∈X^a, s∈I} ⊂X×X^a×I.

Thus Λ_s is a Lagrangian submanifold of X×X^a. Now we can see that Φ is a Hamiltonian isotopy if and only if there exists a Lagrangian submanifold Λ⊂X×X^a×T^∗I such that, for any s∈I,

Λs = Λ◦T_s^∗I.

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(Here, the notation ^• ◦ ^• is a slight generalization of (6) to the case where the symplectic manifolds are no more cotangent bundles.) In this case Λ is written

Λ =

Φ(v, s), v, s,−f(Φ(v, s), s)

;v ∈X, s∈I , (26)

where the function f: X×I −→ R is defined up to addition of a function depending on s by vΦ,s =Hfs.

Homogeneous case

Let us come back to the case X = ˙T^∗M and consider Φ : ˙T^∗M ×I −→T˙^∗M such that

(ϕs is a homogeneous symplectic isomorphism for each s∈I, ϕ0 = idT˙^∗M.

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In this case Φ is a Hamiltonian isotopy and there exists a unique homogeneous function f such that v_Φ,s =H_fs. It is given by

f =hα, v_Φi: ˙T^∗M ×I −→R. (28)

Since f is homogeneous of degree 1 in the fibers of ˙T^∗M, the Lagrangian submanifold Λ of ˙T^∗M ×T˙^∗M ×T^∗I associated tof in (26) is R⁺-conic.

We say that F ∈D(kM) is locally bounded if for any relatively compact open subset U ⊂M we have F|_U ∈D^b(k_U). We denote by D^lb(k_M) the full subcategory of D(kM) consisting of locally bounded objects.

Theorem 2.1. ([2, Th 4.3].) Consider a homogeneous Hamiltonian isotopy Φ satisfying the hypotheses (27). Let us consider the following conditions on K ∈D^lb(kM×M×I):

(a) SS(K)⊂Λ∪T_M^∗_×M×I(M ×M ×I), (b) K0 ≃k∆,

(c) both projections Supp(K)⇒M ×I are proper,

(d) K_s◦K_s⁻¹ ≃ K_s⁻¹◦K_s ≃ k_∆, where K_s⁻¹ = v⁻¹RHom(K_s, ω_M ⊠k_M) and v(x, y) = (y, x).

Then we have

(i) The conditions (a) and (b) imply the other two conditions (c) and (d). (ii) There exists K satisfying (a)–(d).

(iii) Moreover such a K satisfying the conditions (a)–(d) is unique up to a unique isomorphism.

We shall call K the quantization of Φ on I, or the quantization of the family {ϕs}s∈I.

Non homogeneous case

Theorem 2.1 is concerned with homogeneous Hamiltonian isotopies. The next result will allow us to adapt it to non homogeneous cases. Let Φ : T^∗M×I −→ T^∗M be a Hamiltonian isotopy and assume

there exists a compact set C ⊂ T^∗M such that ϕs|_T^∗_M\C is the identity for all s∈I.

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We denote by T_{{τ >0}}^∗ (M×R) the open subset{τ > 0}ofT^∗(M×R) and we define the map

ρ:T_{{τ >0}}^∗ (M ×R)→− T^∗M, (x, t;ξ, τ)7→(x;ξ/τ).

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Proposition 2.2. ([2, Prop. A.6].) There exist a homogeneous Hamiltonian isotopyΦ : ˙e T^∗(M×R)×I −→T˙^∗(M×R) andC^∞-functions u: T^∗M×I −→R and v:I −→R such that the following diagram commutes:

T_{{τ >0}}^∗ (M×R)×I ^{Φe //}

ρ×idI

T_{{τ >0}}^∗ (M ×R)

ρ

T^∗M ×I ^{Φ //}T^∗M and

Φ((x;e ξ),(t;τ), s) = ((x^′;ξ^′),(t+u(x;ξ/τ, s);τ)), (31)

Φ((x;e ξ),(t; 0), s) = ((x;ξ),(t+v(s); 0)), (32)

where (x^′;ξ^′/τ) = ϕ_s(x;ξ/τ). Moreover we have u(x;ξ/τ, s) = v(s) for (x;ξ/τ)6∈C.

3 Convolution and localization

Most of the ideas of this section are due to Tamarkin [12]. The reader will be aware that our notations do not follow Tamarkin’s ones. We also give some proofs which may be rather different from Tamarkin’s original ones.

In all this section, we consider a trivial vector bundle q: E =M ×V −→M

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and a trivial cone γ =M ×γ0 ⊂E such that

γ0 is a closed convex proper cone of V containing 0 and γ0 6={0}.

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The polar cone γ₀^◦ ⊂V^∗ is the closed convex cone given by γ₀^◦ ={θ ∈V^∗;hθ, vi ≥0}for all v ∈γ0.

Many results could be generalized to general vector bundles and general proper convex cones, but in practice we shall use these results with V = R and γ0 ={t ∈R;t≥0}. Recall that a subset in T^∗M ×V^∗ is a cone if it is invariant by the diagonal action of R⁺ (see (17)).

Definition 3.1. A closed cone A ⊂ T^∗M ×V^∗ is called a strict γ-cone if A ⊂(T^∗M ×Intγ₀^◦)∪T_M^∗ M× {0}.

Example 3.2. Assume V = R and M is open in Rⁿ. Denote by (t;τ) the coordinates on T^∗R and by (x;ξ) the coordinates on T^∗M. Let γ0 = {t ∈ R;t ≥ 0}. Then a closed cone A ⊂ T^∗M ×V^∗ is a strict γ-cone if, for any compact subset C ⊂M, there exists a ∈ R, a > 0 such that τ ≥a|ξ| for all (x;ξ, τ)∈A∩(π⁻¹_M(C)×V^∗).

Remark 3.3. Iff: N −→M is a morphism of manifolds andA⊂T^∗M×V^∗ is a strict γ-cone, then f ×idV : N ×V −→ M ×V is non-characteristic for b

π_E⁻¹(A) (where bπ_E⁻¹ is defined in (16)).

In the sequel, we consider the maps

q1, q2, s:V ×V −→V,

q1(v1, v2) =v1, q2(v1, v2) =v2, s(v1, v2) =v1+v2. (35)

If there is no risk of confusion, we still denote byq1, q2, sthe associated maps M ×V ×V −→M ×V.

We denote by δM the diagonal embedding δM: M ֒→M×M (36)

and if there is no risk of confusion, we still denote by δM the associated map M ×V ×V ֒→M ×M ×V ×V, that is, the mapE×_M E ֒→E×E.

The maps s and δM give rise to the maps:

T^∗(E×_M E)←−−−^(δM)d M ×_M×M T^∗(E×_M E)−−−→^(δM)π T^∗(E×E), T^∗(E×M E)←^s−^d V ×V×V T^∗(E×M E)−→^sπ T^∗E.

On T^∗E we have the antipodal map a, but there is another involution asso- ciated with a and the involution (x, y)7→(x,−y) onE. We denote by α the involution of T^∗E

α: (x, y;ξ, η)7→(x,−y;−ξ, η) (37)

and for a subset A ⊂ T^∗E we denote by A^α its image by this involution.

We also denote by α the involution of T^∗M × V^∗ defined by (x;ξ, η) 7→

(x;−ξ, η). Hence for A ⊂ T^∗M × V^∗ we have, using the notation (16), b

π_E⁻¹(A^α) =bπ_E⁻¹(A)^α. Convolution

Recall the notations (10) and (15).

Notation 3.4. For two closed subsets A and B inT^∗E, we set Ab⋆ B :=s♯δ^♯_M(A×B).

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In general, the calculation of Ab⋆ B is difficult. In Lemmas 3.5 and 3.7 below we consider special situations in which this calculation is easy.

Lemma 3.5. LetA^′ andB^′ be two closed cones inV^∗. SetA=T^∗M×V×A^′ and B =T^∗M ×V ×B^′. Then

Ab⋆ B =A∩B.

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Proof. Using the hypothesis on A and B, it follows from (11) that δ^♯_M(A×B) =T^∗M ×V ×V ×A^′×B^′.

Then the result follows from Corollary 1.17.

Notation 3.6. Let A and B be two closed cones in T^∗M ×V^∗. We set A+

MB ={(x;ξ, η)∈T^∗M ×V^∗; there exist ξ1, ξ2 ∈T_x^∗M such that (x;ξ1, η)∈A, (x;ξ2, η)∈B and ξ=ξ1+ξ2}.

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Lemma 3.7. Consider two closed strict γ-cones A and B in T^∗M × V^∗. Then A+

MB is also a strict γ-cone and bπ_E⁻¹(A)b⋆bπ_E⁻¹(B) =πb_E⁻¹(A+

MB).

In particular, if A∩B ⊂T_M^∗ M × {0}, then

(bπ⁻¹_E (A)b⋆(bπ⁻¹_E (B))^α)∩(T_M^∗ M ×T^∗V)⊂ T_E^∗E.

Proof. The fact thatA+

MBis a strictγ-cone follows easily from the definition.

By Remark 3.3, bπ_E⁻¹(A)×bπ⁻¹_E (B) is non-characteristic for the inclusion δM: M ×V ×V −→ M ×M ×V ×V and we may replace δ^♯_M by δM,dδ_M,π⁻¹ in (38). We find δ_M^♯ (bπ_E⁻¹(A)×(πb_E⁻¹(B))) =bπ⁻¹_M×V×V(C1), where

C₁ ={(x;ξ, η₁, η₂)∈T^∗M ×V^∗×V^∗; there exist ξ₁, ξ₂ ∈T_x^∗M such that (x;ξ1, η1)∈A, (x;ξ2, η2)∈B and ξ=ξ1+ξ2} and the result follows.

Using the notations (35), the convolution of sheaves is defined by:

Definition 3.8. For F, G∈D^b(kE), we set

F ⋆ G := Rs_!(q⁻¹₁ F ⊗^L q₂⁻¹G)≃Rs_!δ_M⁻¹(F ⊠^L G), (41)

F ⋆npG := Rs∗(q⁻¹₁ F ⊗^L q₂⁻¹G)≃Rs∗δ⁻¹_M(F ⊠^L G).

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The morphism kγ −→kM×{0} gives the morphism F ⋆npkγ −→F.

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Recall the following result:

Proposition 3.9. (Microlocal cut-off lemma [5, Prop. 5.2.3, 3.5.4]) LetF ∈ D^b(kE). Then SS(F) ⊂T^∗M ×V ×γ₀^◦ if and only if the morphism (43) is an isomorphism.

If γ₀ has a non-empty interior we have k_γ0 ≃ D^′_V(k_Intγ0) and we deduce from Corollary 1.7 (ii) that

F ⋆_npk_γ ≃Rs_∗RΓ_M×V×Intγ0(q₁⁻¹F).

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Following Tamarkin [12], we introduce a right adjoint to the convolution functor by setting for F, G∈D^b(kE)

Hom^∗(G, F) := Rq1∗RHom(q₂⁻¹G, s^!F).

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Hence for F1, F2, F3 ∈D^b(kE), we have

RHom (F1⋆ F2, F3)≃RHom (F1,Hom^∗(F2, F3)).

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We use the notation:

i: E −→E denotes the involution (x, y)7→(x,−y).

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