Microlocal Euler classes and Hochschild homology

arXiv:1203.4869v4 [math.AG] 1 May 2013

Microlocal Euler classes and Hochschild homology

Masaki Kashiwara

and Pierre Schapira May 2, 2013

Abstract

We define the notion of a trace kernel on a manifold M. Roughly speaking, it is a sheaf onM×M for which the formalism of Hochschild homology applies. We associate a microlocal Euler class to such a ker- nel, a cohomology class with values in the relative dualizing complex of the cotangent bundleT^∗M over M and we prove that this class is functorial with respect to the composition of kernels.

This generalizes, unifies and simplifies various results of (relative) index theorems for constructible sheaves,D-modules and elliptic pairs.

Contents

1 Introduction 2

2 A short review on sheaves 5

3 Compositions of kernels 7

4 Microlocal homology 9

5 Microlocal Euler classes of trace kernels 16

6 Operations on microlocal Euler classes I 24

∗This work was partially supported by Grant-in-Aid for Scientific Research (B) 22340005, Japan Society for the Promotion of Science.

7 Operations on microlocal Euler classes II 27 8 Applications: D-modules and elliptic pairs 34

9 The Lefschetz fixed point formula 36

1 Introduction

Our constructions mainly concern real manifolds, but in order to introduce the subject we first consider a complex manifold (X,O_X). Denote byω_X^holthe dualizing complex in the category of O_X-modules, that is, ω^hol_X = ΩX[dX], whered_X is the complex dimension of X and ΩX is the sheaf of holomorphic forms of degree d_X. Denote by O_∆

X and ω_∆^hol_X the direct images of O_X and ω_X^holrespectively by the diagonal embeddingδ: X ֒→ X×X. It is well-known (see in particular [Ca05,CaW07]) that the Hochschild homology of O_X may be defined by using the isomorphism

δ∗HH (O_X)≃RHomO_X×X O_∆

X, ω_∆^hol_X (1.1) .

Moreover, if F is a coherent O_X-module and D^OF := RHom_O

X(F, ω_X^hol) denotes its dual, there are natural morphisms

O_∆

X −→F ⊠D^OF −→ω_∆^hol_X (1.2)

whose composition defines the Hochschild class of F: hh^O(F)∈H_Supp(⁰ F)(X;HH(O_X)).

These constructions have been extended when replacing O_X with a so-called DQ-algebroid stackA_X in [KS12] (DQ stands for “deformation-quantization”).

One of the main results of loc. cit. is that Hochschild classes are functorial with respect to the composition of kernels, a kind of (relative) index theorem for coherent DQ-modules.

On the other hand, the notion of Lagrangian cycles of constructible sheaves on real analytic manifolds has been introduced by the first named author (see [Ka85]) in order to prove an index theorem for such sheaves, after they first appeared in the complex case (see [Ka73] and [McP74]). We refer to [KS90, Chap. 9] for a systematic study of Lagrangian cycles and for historical comments. Let us briefly recall the construction.

Consider a real analytic manifold M and let k be a unital commutative ring with finite global dimension. Denote by ω_M the (topological) dualizing complex of M, that is, ω_M = orM[dimM] where orM is the orientation sheaf of M and dimM is the dimension. Finally, denote by πM: T^∗M −→ M the cotangent bundle to M. Let Λ be a conic subanalytic Lagrangian subset of T^∗M. The group of Lagrangian cycles supported by Λ is given by H_Λ⁰(T^∗M;π⁻_M¹ωM). Denote by D^bR-c(kM) the bounded derived category ofR- constructible sheaves on M. To an object F of this category, one associates a Lagrangian cycle supported by SS(F), the microsupport ofF. This cycle is called the characteristic cycle, or the Lagrangian cycle or else the microlocal Euler class of F and is denoted here by µeu_M(F).

In fact, it is possible to treat the microlocal Euler classes ofR-constructi- ble sheaves on real manifolds similarly as the Hochschild class of coherent sheaves on complex manifolds. Denote as above by k_∆M and ω∆M the direct image of k_M and ω_M by the diagonal embedding δ_M: M ֒→ M ×M. Then we have an isomorphism

H_Λ⁰(T^∗M;π_M⁻¹ωM)≃H_Λ⁰ T^∗M;µhom(k∆M, ω∆M) (1.3) ,

where µhomis the microlocalization of the functor RHom. Thenµeu_M(F) is obtained as follows. Denote by DMF := RHom(F, ωM) the dual of F. There are natural morphisms

k_∆M −→F ⊠DMF −→ω_∆M, (1.4)

whose composition gives the microlocal Euler class of F.

In this paper, we construct the microlocal Euler class for a wide class of sheaves, including of course the constructible sheaves but also the sheaves of holomorphic solutions of coherent D-modules and, more generally, of ellip- tic pairs in the sense of [ScSn94]. To treat such situations, we are led to introduce the notion of a trace kernel.

On a real manifold M (say of class C^∞), a trace kernel is the data of a triplet (K, u, v) where K is an object of the derived category of sheaves D^b(k_M×M) and u, v are morphisms

u: k_∆M −→K, v: K −→ω_∆M. (1.5)

One then naturally defines the microlocal Euler class µeu_M(K, u, v) of such a kernel, an element of H_Λ⁰(T^∗M;µhom(k∆M, ω∆M)) where Λ = SS(K)∩ T_∆^∗_M(M ×M). By (1.4), a constructible sheaf gives rise to a trace kernel.

IfXis a complex manifold andM is a coherentD_X-module, we construct natural morphisms (over the base ring k=C)

C_∆

X −→ΩX×X

⊗L_D

X×X(M⊠DDM)−→ ω_∆X, (1.6)

where D_DM denotes the dual of M as a D-module. In other words, one naturally associates a trace kernel onX to a coherentD_X-module. Moreover, we prove that under suitable microlocal conditions, the tensor product of two trace kernels is again a trace kernel, and it follows that one can associate a trace kernel to an elliptic pair.

We study trace kernels and their microlocal Euler classes, showing that some proofs of [KS12] can be easily adapted to this situation. One of our main results is the functoriality of the microlocal Euler classes: the microlocal Euler class of the composition K₁◦K₂ of two trace kernels is the composition of the microlocal Euler classes of K₁ and K₂ (see Theorem 6.3 for a precise statement). Another essential result (which is far from obvious) is that the composition of classes coincides with the composition forπ_M⁻¹ωM constructed in [KS90] via the isomorphism between µhom(k_∆M, ω_∆M) and π_M⁻¹ω_M.

As an application, we recover in a single proof the classical results on the index theorem for constructible sheaves (see [KS90,§ 9.5]) as well as the index theorem for elliptic pairs of [ScSn94], that is, sheaves of generalized holomorphic solutions of coherent D-modules. We also briefly explain how to adapt trace kernels to the formalism of the Lefschetz trace formula.

We call here µhom(k_∆M, ω_∆M) the microlocal homology of M, and this paper shows that, in some sense, the microlocal homology of real manifolds plays the same role as the Hochschild homology of complex manifolds.

To conclude this introduction, let us make a general remark. The cate- gory D^bR-c(kM) of constructible sheaves on a compact real analytic manifold M is “proper” in the sense of Kontsevich (that is, Ext finite) but it does not admit a Serre functor (in the sense of Bondal-Kapranov) and it is not clear whether it is smooth (again in the sense of Kontsevich). However this category naturally appears in Mirror Symmetry (see [FLTZ10]) and it would be a natural question to try to understand its Hochschild homology in the sense of [McC94, Ke99]. We don’t know how to compute it, but the above construction, with the use of µhom(k∆M, ω_∆M), provides an alternative ap- proach of the Hochschild homology of this category. This result is not totally surprising if one remembers the formula (see [KS90, Prop. 8.4.14]):

DT^∗M(µhom(F, G))≃µhom(G, F)⊗π_M⁻¹ωM.

Hence, in some sense, π⁻_M¹ωM plays the role of a microlocal Serre functor.

Note that thanks to Nadler and Zaslow [NZ09], the category D^bR-c(kM) is equivalent to the Fukaya category of the symplectic manifold T^∗M, and this is another argument to treat sheaves from a microlocal point of view.

Acknowledgments

The second named author warmly thanks St´ephane Guillermou for helpful discussions.

2 A short review on sheaves

Throughout this paper, a manifold means a real manifold of class C^∞. We shall mainly follow the notations of [KS90] and use some of the main notions introduced there, in particular that of microsupport and the functor µhom.

LetM be a manifold. We denote byπM: T^∗M −→M its cotangent bundle.

For a submanifold N of M, we denote by T_N^∗M the conormal bundle to N. In particular, T_M^∗ M denotes the zero-section. We set ˙T^∗M :=T^∗M \T_M^∗M and we denote by ˙πM the restriction of πM to ˙T^∗M. If there is no risk of confusion, we write simply π and ˙π instead of π_M and ˙π_M. One denotes by a: T^∗M −→ T^∗M the antipodal map, (x;ξ)7→ (x;−ξ) and for a subset S of T^∗M, one denotes by S^a its image by this map. A set A ⊂ T^∗M is conic if it is invariant by the action of R⁺ onT^∗M.

Letf: M −→N be a morphism of manifolds. Tof one associates as usual the maps

T^∗M

πM

))

❙❙

❙❙

❙❙

❙❙

❙❙

❙❙

❙❙

❙❙

❙ M ×N T^∗N

π

fd

oo ^fπ //T^∗N

πN

M ^{f //}N.

(2.1)

(Note that in loc. cit. the map f_d is denoted by ^tf^′−1.)

Let Λ be a closed conic subset ofT^∗N. One says thatfisnon-characteris- ticfor Λ if the mapfdis proper onf_π⁻¹Λ or, equivalently,f_π⁻¹Λ∩f_d⁻¹(T_M^∗ M)⊂ M ×N T_N^∗N.

Let k be a commutative unital ring with finite global homological di- mension. One denotes by k_M the constant sheaf on M with stalk k and by D^b(k_M) the bounded derived category of sheaves of k-modules onM. When M is a real analytic manifold, one denotes byD^bR-c(kM) the full triangulated subcategory of D^b(kM) consisting of R-constructible objects.

One denotes byωM the dualizing complex onM and byω_M^⊗−1its dual, that is, ω_M^⊗−1 = RHom(ωM,k_M). More generally, for a morphism f: M −→ N, one denotes by ω_M/N := f^!k_N ≃ ω_M ⊗f⁻¹(ω_N^⊗−1) the relative dualizing complex. Recall that ωM ≃orM[dimM] where orM is the orientation sheaf and dimM is the dimension of M. Also recall the natural morphism of functors

ωM/N ⊗f⁻¹ −→f^!. (2.2)

We have the duality functors

D^′_MF = RHom(F,k_M), D_MF = RHom(F, ω_M).

For F ∈ D^b(kM), one denotes by Supp(F) the support of F and by SS(F) its microsupport, a closedR⁺-conic co-isotropic subset ofT^∗M. For a morphism f: M −→N andG∈D^b(k_N), one says thatf is non-characteristic for G if f is non-characteristic for SS(G).

We shall use systematically the functorµhom, a variant of Sato’s microlo- calization functor. Recall that for a closed submanifold N of M, there is a functor µN: D^b(kM) −→ D^b(kT_N^∗M) constructed by Sato (see [SKK73]) and for F₁, F₂ ∈D^b(kM), one defines in [KS90] the functor

µhom:D^b(kM)^op×D^b(kM)−→D^b(kT^∗M),

µhom(F1, F2) :=µ∆RHom(q₂⁻¹F1, q₁^!F2)

where q₁ and q₂ are the first and second projection defined on M ×M and

∆ is the diagonal. This sheaf is supported by T_∆^∗(M ×M) that we identify with T^∗M by the first projectionT^∗(M×M) ≃T^∗M×T^∗M −→T^∗M. Note that

Supp(µhom(F1, F2))⊂SS(F1)∩SS(F2) (2.3)

and we have Sato’s distinguished triangle, functorial in F₁ and F₂: Rπ_!µhom(F₁, F₂)−→Rπ∗µhom(F₁, F₂)−→R ˙π∗ µhom(F₁, F₂)|T˙^∗M

+1

−→. (2.4)

Moreover, we have the isomorphism

Rπ^∗µhom(F1, F₂)≃RHom(F1, F₂), (2.5)

and, assuming thatM is real analytic andF1 is R-constructible, the isomor- phism

Rπ!µhom(F1, F2)≃D^′_MF1

⊗L F2. (2.6)

In particular, assuming that F1 is R-constructible and SS(F1)∩SS(F2) ⊂ T_M^∗ M, we have the natural isomorphism (see [KS90, Cor 6.4.3])

D^′_MF1

⊗L F2 −→∼ RHom(F1, F2).

(2.7)

As recalled in the Introduction, assuming that M is real analytic, we have the formula (see [KS90, Prop. 8.4.14]):

DT^∗M(µhom(F1, F₂))≃µhom(F2, F₁)⊗π_M⁻¹ω_M for F₁, F₂ ∈D^bR-c(kM).

(2.8)

3 Compositions of kernels

Notation 3.1. (i) For a manifold M, let δ_M: M −→ M ×M denote the diagonal embedding, and ∆M the diagonal set of M ×M.

(ii) Let Mi (i= 1,2,3) be manifolds. For short, we write Mij :=Mi×Mj

(1≤ i, j ≤ 3), M₁₂₃ =M₁ ×M₂×M₃, M₁₂₂₃ = M₁×M₂×M₂ ×M₃, etc.

(iii) We will often write for shortk_i instead of k_Mi and k_∆i instead of k_∆Mi and similarly withωMi, etc., and with the indexireplaced with several indicesij, etc.

(iv) We denote by π_i, π_ij, etc. the projection T^∗M_i →− M_i, T^∗M_ij −→ M_ij, etc.

(v) We denote by qi the projectionMij −→Mi or the projectionM123 −→Mi

and by q_ij the projection M₁₂₃ −→ M_ij. Similarly, we denote by p_i the projection T^∗Mij −→ T^∗Mi or the projection T^∗M123 −→ T^∗Mi and by pij the projection T^∗M123 −→T^∗Mij.

(vi) We also need to introduce the maps pj^a or pij^a, the composition of pj

orp_ij and the antipodal map onT^∗M_j. For example, p₁₂^a((x₁, x₂, x₃;ξ₁, ξ₂, ξ₃)) = (x₁, x₂;ξ₁,−ξ₂).

(vii) We let δ2: M123 −→M1223 be the natural diagonal embedding.

We consider the operation of composition of kernels:

◦2 : D^b(k_M12)×D^b(k_M23)−→D^b(k_M13) (K1, K2)7→K1◦

2K2 := Rq_13!(q₁₂⁻¹K1

⊗L q⁻₂₃¹K2)

≃ Rq_13!δ⁻₂¹(K1 L

⊠K2).

(3.1)

We will use a variant of ◦:

∗2 : D^b(k_M12)×D^b(k_M23)−→D^b(k_M13) (K1, K₂)7→K₁∗

2K₂:= Rq13∗ q₂⁻¹ω₂⊗δ₂^!(K1 L

⊠K₂) (3.2) .

We also have ω_M123/M1223 ≃ q₂⁻¹ω_M^⊗−₂¹ and we deduce from (2.2) a morphism δ₂⁻¹ −→q₂⁻¹ωM2⊗δ₂^! . Using the morphism Rp_13! −→Rp13∗ we obtain a natural morphism for K₁ ∈D^b(kM12) andK₂ ∈D^b(kM23):

K1◦K2 −→K1∗K2. (3.3)

It is an isomorphism if p⁻₁₂^1aSS(K1)∩p₂₃^−1aSS(K2)−→T^∗M13 is proper.

We define the composition of kernels on cotangent bundles (see [KS90, Prop. 4.4.11])

◦a

2 : D^b(k_T^∗_M12)×D^b(k_T^∗_M23) −→ D^b(k_T^∗_M13) (K1, K2) 7→ K1

◦a

2K2:= Rp_13!(p⁻₁₂^1aK1

⊗L p⁻₂₃¹K2)

≃Rp13^a!(p⁻₁₂^1aK₁⊗^L p⁻₂₃¹aK₂).

(3.4)

We also define the corresponding operations for subsets of cotangent bundles.

Let A⊂T^∗M12 and B ⊂T^∗M23. We set A×^a

2 B =p⁻₁₂^1a(A)∩p⁻₂₃¹(B), A◦^a

2B =p13(A×^a

2 B)

=

(x₁, x₃;ξ₁, ξ₃)∈T^∗M₁₃; there exists (x₂;ξ₂)∈T^∗M₂ such that (x1, x₂;ξ₁,−ξ2)∈A,(x2, x₃;ξ₂, ξ₃)∈B

. (3.5)

We have the following result which slightly strengthens Proposition 4.4.11 of [KS90] in which the composition ∗ is not used.

Proposition 3.2. For G1, F1 ∈ D^b(kM12) and G2, F2 ∈ D^b(kM23) there ex- ists a canonical morphism (whose construction is similar to that of [KS90, Prop. 4.4.11] ):

µhom(G1, F₁)◦^a

2µhom(G2, F₂)−→µhom(G1∗

2G₂, F₁◦

2F₂).

Proof. In Proposition 4.4.8 (i) of loc. cit., one may replace F₂ ⊠^LS G₂ with j^!(F₂⊠^L G₂)⊗ω_X^⊗−_×¹

SY /X×Y. Then the proof goes exactly as that of Proposi-

tion 4.4.11 in loc. cit. Q.E.D.

Let Λ_ij ⊂ T^∗M_ij (i= 1,2, j =i+ 1) be closed conic subsets and consider the condition:

the projectionp₁₃: Λ₁₂×^a

2 Λ₂₃−−→T^∗M₁₃ is proper.

(3.6) We set

Λ13= Λ12

◦a 2Λ23. (3.7)

Corollary 3.3. Assume that Λij (i= 1,2, j =i+ 1) satisfy (3.6). We have a composition morphism

RΓΛ12µhom(G1, F₁)◦^a

2RΓΛ23µhom(G2, F₂)−→RΓΛ13µhom(G1∗

2G₂, F₁◦

2F₂).

Convention 3.4. In (3.1), we have introduced the composition ◦

2 of kernels K₁ ∈ D^b(kM12) and K₂ ∈ D^b(kM23). However we shall also use the nota- tion M₂₂ = M₂×M₂ and consider for example kernels L₁ ∈ D^b(k_M122) and L2 ∈ D^b(kM223). Then when writing L1◦

2L2 we mean that the composition is taken with respect to the last variable of M₂₂ for L₁ and the first variable for L2. In other words, set M4 =M2 and consider L1 and L2 as objects of D^b(kM142) and D^b(kM243) respectively, in which case the composition L₁◦

2L₂ is unambiguously defined.

4 Microlocal homology

LetM be a real manifold. Recall thatδM: M ֒→M×M denotes the diagonal embedding. We shall identify M with the diagonal ∆M of M ×M and we

sometimes write ∆ instead of ∆M if there is no risk of confusion. We shall identify T^∗M with T_∆^∗(M ×M) by the map

δ_T^a∗_M: T^∗M ^{ //}T^∗(M ×M), (x;ξ)7→(x, x;ξ,−ξ).

We denote by k_∆M, ω∆M and ω_∆^⊗−_M¹ the direct image by δM of k_M, ωM and ω_M^⊗−1:= RHom(ωM,k_M), respectively.

The next definition is inspired by that of Hochschild homology on complex manifolds (see Introduction).

Definition 4.1. Let Λ be a closed conic subset of T^∗M. We set MH_Λ(kM) := RΓΛ(δ^a_T^∗_M)⁻¹µhom(k∆M, ω∆M),

MH_Λ(kM) := RΓ(T^∗M;MH_Λ(kM)), MH^k

Λ(kM) := H^k(MH_Λ(kM)) =H^k(T^∗M;MH_Λ(kM)).

(4.1)

We call MH_Λ(k_M) themicrolocal homology of M with support in Λ.

We also write MH(kM) instead of MH_T^∗_M(kM).

Remark 4.2. (i) We have µhom(k∆M, ω_∆M) ≃ (δ_T^a∗_M)∗π_M⁻¹ω_M. In par- ticular, we have MH_Λ(k_M) ≃ RΓ_Λ(T^∗M;π_M⁻¹ω_M) and MH(k_M) ≃ RΓ(M;ωM). Assuming that M is real analytic and Λ is a closed conic subanalytic Lagrangian subset of T^∗M, we recover the space of La- grangian cycles with support in Λ as defined in [KS90, §9.3].

(ii) The support of µhom(k∆M, ω_∆M) is T_∆^∗_M(M ×M). Hence, we have RΓ_δ^a

T∗MΛµhom(k_∆M, ω_∆M)≃(δ_T^a∗_M)∗MH_Λ(k_M).

(iii) IfM is real analytic and Λ is a Lagrangian subanalytic closed conic sub- set, then we haveH^k(MH_Λ(k_M)) = 0 fork <0 (see [KS90, Prop. 9.2.2]).

In the sequel, we denote by ∆i (resp. ∆ij) the diagonal subset ∆Mi ⊂Mii

(resp. ∆Mij ⊂M_iijj).

Lemma 4.3. We have natural morphisms:

(i) ω∆12 ◦

22(k∆2

L

⊠ω∆3)−→ω∆13, (ii) k_∆13 −→k_∆12 ∗

22(ω^⊗−_∆2¹⊠^L k_∆3).

Proof. Denote by δ22 the diagonal embedding M112233 ֒→M11222233. (i) We have the morphisms

ω_∆12 ◦

22(k∆2

L

⊠ω_∆3) = Rq_1133!δ₂₂⁻¹(ω∆12

L

⊠k_∆2

L

⊠ω_∆3)

≃ Rq_1133!ω_∆123

−

→ ω_∆13. (ii) The isomorphism

δ₂₂^! (k∆2 ⊠ω∆2)≃k_∆2

gives rise to the isomorphisms k_∆12 ∗

22(ω_∆^⊗−₂¹⊠^L k_∆3) = Rq₁₁₃₃∗ q⁻₁₁₃₃¹ ω₂₂⊗δ₂₂^! (k_∆12

L

⊠ω^⊗−_∆2¹⊠^L k_∆3)

≃ Rq1133∗δ₂₂^! (k∆1

L

⊠ω_∆2

L

⊠k_∆23)

≃ Rq1133∗k_∆123

and the result follows by adjunction from the morphism

q₁₁₃₃⁻¹ k_∆13 ≃k_∆1 ⊠^L k₂₂⊠^L k_∆3 −→k_∆1 ⊠^L k_∆2 ⊠^L k_∆3 =k_∆123.

Q.E.D.

Proposition 4.4. Let M_i (i = 1,2,3) be manifolds. We have a natural composition morphism (whose constructions will be given in the course of the proof):

µhom(k_∆12, ω_∆12)◦^a

22µhom(k_∆23, ω_∆23)−→µhom(k_∆13, ω_∆13).

(4.2)

In particular, let Λij be a closed conic subset of T^∗Mij (ij = 12,13,23). If Λ12

◦a

2Λ23⊂ Λ13, then we have a morphism MH_Λ12(k12) ◦^a

2

MH_Λ23(k23)−→MH_Λ13(k13).

(4.3)

Proof. Consider the morphism (see Proposition 3.2 and Convention 3.4) µhom(ω_∆^⊗−₂¹, ω_∆^⊗−₂¹)◦^a

2µhom(k∆23, ω_∆23) −→ µhom(ω_∆^⊗−₂¹∗

2k_∆23, ω_∆^⊗−₂¹◦

2ω_∆23)

≃ µhom(ω_∆^⊗−₂¹⊠^L k_∆3,k_∆2

L

⊠ω∆3).

It induces an isomorphism

µhom(k∆23, ω∆23)≃µhom(ω_∆^⊗−₂¹⊠^L k_∆3,k_∆2

L

⊠ω∆3).

(4.4)

Note that this isomorphism is also obtained by

µhom(k∆23, ω_∆23) ≃ µhom (ω^⊗−₂ ¹⊠^L k₂₃₃)⊗^L k_∆23,(ω₂^⊗−1⊠^L k₂₃₃)⊗^L ω_∆23

≃ µhom(ω_∆^⊗−₂¹⊠^L k_∆3,k_∆2 ⊠^L ω_∆3).

Applying Proposition 3.2, we get a morphism:

µhom(k∆12, ω∆12)◦^a

22µhom(k∆23, ω∆23)

−

→µhom(k∆12 ∗

22(ω_∆^⊗−₂¹⊠^L k_∆3), ω∆12 ◦

22(k∆2

L

⊠ω∆3)).

(4.5)

It remains to apply Lemma 4.3. Q.E.D.

Corollary 4.5. Let Λij (i = 1,2, j = i+ 1) satisfying (3.6) and let Λ13 = Λ12

◦a

2Λ23. The composition of kernels in (4.3) induces a morphism

◦a

2 : MH_Λ12(k12) ⊗^L MH_Λ23(k23)−→MH_Λ13(k13).

(4.6)

In particular, each λ∈ MH⁰

Λ12(k₁₂) defines a morphism λ◦^a

2 : MH_Λ23(k23)−→MH_Λ13(k13).

(4.7)

Proof. These morphisms follow from (4.3).The second assertion follows from the isomorphism H⁰(X)≃Hom_D^b_(k)(k, X) in the category D^b(k). Q.E.D.

Theorem 4.6. (i) We have the isomorphisms

µhom(k_∆M, ω_∆M) ≃ (δ_T^a∗_M)∗π⁻_M¹RHom(k_M, ω_M)

≃ (δ_T^a∗_M)∗π⁻_M¹ω_M. (ii) We have a commutative diagram

µhom(k∆12, ω∆12)◦^a

22µhom(k∆23, ω∆23) ^//

≀

µhom(k∆13, ω∆13)

≀

(δ_T^a∗_M13)∗ π⁻_M¹₁₂ω_M12

◦a

2π⁻_M¹₂₃ω_M23

//(δ_T^a∗_M13)∗π_M⁻¹₁₃ω_M13. (4.8)

Here the top horizontal arrow of (4.8) is given in Proposition4.4, and the bottom horizontal arrow is induced by

p⁻₁₂^1aπ⁻_M¹₁₂ωM12

⊗L p⁻₂₃¹π_M⁻¹_23aωM23 ≃π_M⁻¹₁ωM1

L

⊠π_M⁻¹₂(ωM2

⊗L ωM2)⊠^L π_M⁻¹₃ωM3, π⁻_M¹₂(ω_M2

⊗L ω_M2)≃ω_T^∗_M2, Rp_13! π_M⁻¹₁ω_M1

L

⊠ω_T^∗_M2

L

⊠π_M⁻¹₃ω_M3

−−→π⁻_M¹₁ω_M1

L

⊠π⁻_M¹₃ω_M3. Proof. (i) is obvious.

(ii)–(a) By [KS90, Prop. 4.4.8], we have natural morphisms for (i, j) = (1,2) or (i, j) = (2,3):

µhom(k_∆i, ω_∆i)⊠^L µhom(k_∆j, ω_∆j) −→ µhom(k_∆ij, ω_∆ij)

and it follows from (i) that these morphisms are isomorphisms. These iso- morphisms give rise to the isomorphism

µhom(k_∆12, ω_∆12)◦^a

22µhom(k_∆23, ω_∆23) ≃ µhom(k∆1, ω∆1)⊠^L µhom(k∆2, ω∆2)◦^a

22µhom(k∆2, ω∆2) ^L

⊠µhom(k∆3, ω∆3) Similarly, we have an isomorphism

π⁻_M¹₁₂ω_M12

◦a

2π⁻_M¹₂₃ω_M23 ≃ π_M⁻¹₁ω_M1 ⊠ π_M⁻¹₂ω_M2

◦a

2π_M⁻¹₂ω_M2

⊠π_M⁻¹₁ω_M1.

Hence, we are reduced to the case where M₁ = M₃ = pt, which we shall assume now.

(ii)–(b) We change our notations and we set:

M :=M2, Y :=M ×M,

δM: M ֒→Y the diagonal embedding,∆M =δM(M), j: Y ֒→Y ×Y the diagonal embedding, ∆_Y =δ_Y(Y), δ^a_T^∗_M: T^∗M ֒→T^∗Y, (x;ξ)7→(x, x;ξ,−ξ),

δ^a_T^∗_Y : T^∗Y ֒→T^∗Y ×T^∗Y, p: T^∗Y −→pt the projection, aY : Y −→pt the projection.

With these new notations, the composition ◦^a

22 will be denoted by ◦^a

T^∗Y. Consider the diagram4.9 similar to Diagram (4.4.15) of [KS90]

T^∗M ×T^∗M ^{ i //}T^∗Y ×T^∗Y ^{δ ? _}T^∗Y

a T∗Y

o

p

""

❉❉

❉❉

❉❉

❉❉

❉❉

❉❉

❉❉

❉❉

❉❉

❉❉

❉❉

T^∗Y ×Y T^∗Y

jπ

OO

jd

T_∆^∗_Y(Y ×Y)

e ? _

o s

πY

p1

∼

OO

T^∗Y

s Y

oo _a

Y //pt.

(4.9)

Here, iis the canonical embedding induced byδ_T^a∗_M,p1 is induced by the first projection T^∗Y ×T^∗Y −→ T^∗Y, s: Y ֒→T^∗Y is the zero-section embedding and es is the natural embedding. Note that the square labelled by is Cartesian. We have

Rp! ◦(δ^a_T^∗_Y)⁻¹ ≃ RaY! ◦RπY! ◦p⁻₁¹◦(δ^a_T^∗_Y)⁻¹

≃ RaY! ◦RπY! ◦es⁻¹ ◦j_π⁻¹

≃ RaY! ◦s⁻¹◦Rjd! ◦j_π⁻¹. Therefore,

µhom(k_∆M, ω_∆M) ◦^a

T^∗Y µhom(k_∆M, ω_∆M)

≃Rp!(δ_T^a∗_Y)⁻¹ µhom(k∆M, ω∆M)⊠^L µhom(k∆M, ω∆M)

≃RaY!s⁻¹Rjd!j_π⁻¹µhom(k∆M

L

⊠k_∆M, ω∆M

L

⊠ω∆M).

Hence, by adjunction, to give a morphism µhom(k∆M, ω∆M) ◦^a

T^∗Y µhom(k∆M, ω∆M)−→k is equivalent to giving a morphism in D^b(k_Y)

s⁻¹Rjd!j_π⁻¹µhom(k∆M

L

⊠k_∆M, ω∆M

L

⊠ω∆M)−→a_Y^! k_pt. (4.10)

Note that the left hand side of (4.10) is supported on ∆M. Hence in order to give a morphism (4.10), it is necessary and sufficient to give a morphism in D^b(kM)

δM

−1s⁻¹Rjd!j_π⁻¹µhom(k∆M

L

⊠k_∆M, ω∆M

L

⊠ω∆M)−→δM!a_Y^! k_pt. (4.11)

Hence, it is enough to check the commutativity of the upper square in the following diagram in D^b(kM)

δM

−1s⁻¹Rjd!j_π⁻¹µhom(k∆M

L

⊠k_∆M, ω∆M

L

⊠ω∆M) ^//

∼

δ_M^! a_Y^! k_pt

id

δM−1s⁻¹Rjd!j_π⁻¹i∗ π⁻_M¹ωM L

⊠π_M⁻¹ωM

//

∼

δ_M^! a_Y^! k_pt

∼

ωM id

//ωM.

(4.12)

The top horizontal arrow is constructed by the chain of morphisms (see [KS90,

§ 4.4]):

Rjd!j_π⁻¹µhom(k∆M

L

⊠k_∆M, ω∆M

L

⊠ω∆M)

−

→µhom(j^!(k_∆M ⊠^L k_∆M)⊗^L ω_Y, j⁻¹(ω_∆M ⊠^L ω_∆M))

≃µhom(ω∆M, ω∆M ⊗ω∆M)≃(δ_T^a∗_M)_∗π_M⁻¹ωM

and

δ_M⁻¹s⁻¹Rjd!j_π⁻¹µhom(k_∆M ⊠^L k_∆M, ω_∆M ⊠^L ω_∆M)

−

→δ_M⁻¹s⁻¹(δ_T^a∗_M)∗π_M⁻¹ω_M ≃ω_M. (4.13)

Hence, the commutativity of the diagram (4.12) is reduced to the commuta- tivity of the diagram below:

δ_M⁻¹s⁻¹Rjd!j_π⁻¹µhom(k∆M

L

⊠k_∆M, ω_∆M ⊠^L ω_∆M)

λ

++

❱❱

❱❱

❱❱

❱❱

❱❱

❱❱

❱❱

❱❱

❱❱

❱❱

❱❱

❱❱

❱❱

❱❱

❱❱

δ_M⁻¹s⁻¹Rjd!j_π⁻¹i∗ π_M⁻¹ω_M ⊠^L π⁻_M¹ω_M ∼

//ω_M.

(4.14)

where the morphism λ is given by the morphisms in (4.13). All terms of (4.14) are concentrated at the degree −dimM. Hence the commutativity of (4.14) is a local problem in M and we can assume that M is a Euclidean space. We can checked directly in this case. Q.E.D.

Remark 4.7. Theorem 4.6 may be applied as follows. Let Λij be a closed conic subset of T^∗M_ij (i = 1.2, j = i+ 1). Assume (3.6), that is, the projection p₁₃: Λ₁₂×^a

2 Λ₂₃−−→T^∗M₁₃ is proper and set Λ₁₃ = Λ₁₂◦^a

2Λ₂₃. Let λij ∈MH⁰

Λij(kMij)≃H_Λ⁰_ij(T^∗Mij;π⁻¹ωij). Then λ12

◦a 2λ23=

Z

T^∗M2

λ12∪λ23

(4.15)

where the right hand-side is obtained as follows. Set Λ := Λ₁₂×^a

2 Λ₂₃ and consider the morphisms

H_Λ⁰₁₂(T^∗M12;π⁻¹ω12)×H_Λ⁰₂₃(T^∗M23;π⁻¹ω23)

−

→H_Λ⁰(T^∗M₁₂₃;π⁻¹ω₁⊠^L ω_T^∗_M2

L

⊠π⁻¹ω₃)

−

→H_Λ⁰₁₃(T^∗M₁₃;π⁻¹ω₁₃).

The first morphism is the cup product and the second one is the integration morphism with respect to T^∗M2.

5 Microlocal Euler classes of trace kernels

In this section, we often write ∆ instead of ∆_M.

Definition 5.1. Atrace kernel(K, u, v) onM is the data ofK ∈D^b(kM×M) together with morphisms

k_∆ −−→^u K and K −−→^v ω_∆. (5.1)

In the sequel, as far as there is no risk of confusion, we simply write K instead of (K, u, v).

For a trace kernelK as above, we set

SS∆(K) := SS(K)∩T_∆^∗(M ×M) = (δ_T^a∗_M)⁻¹SS(K).

(5.2)

(Recall that one often identifies T^∗M and T_∆^∗(M ×M) by δ_T^a∗_M: T^∗M ֒→ T^∗M ×T^∗M.)

Definition 5.2. Let (K, u, v) be a trace kernel.

(a) The morphism u defines an element ˜u in H_SS⁰ _∆(K)(T^∗M;µhom(k∆, K)) and the microlocal Euler class µeu_M(K) of K is the image of ˜u by the morphism µhom(k∆, K)−→µhom(k∆, ω∆) associated with the mor- phism v.

(b) Let Λ be a closed conic subset of T^∗M containing SS∆(K). One denotes by µeu_Λ(K) the image of ˜u in H_Λ⁰ T^∗M;µhom(k∆, ω_∆)

. Hence,

µeu_Λ(K)∈MH⁰

Λ(kM)≃H_Λ⁰(T^∗M;π⁻¹ωM).

(5.3)

Let ˜v be the element ofH_SS⁰ _∆(K)(T^∗M;µhom(K, ω∆)) induced byv. Then the microlocal Euler classµeu_M(K) ofK coincides with the image of ˜v by the morphism µhom(K, ω∆M) −→ µhom(k∆, ω∆) associated with the morphism u, which can be easily seen by the commutative diagram:

(δ_T^a∗_M)⁻¹µhom(K, K) ^{v //}

u

(δ_T^a∗_M)⁻¹µhom(K, ω∆)

u

(δ^a_T^∗_M)⁻¹µhom(k_∆, K) ^{v //}(δ^a_T^∗_M)⁻¹µhom(k_∆, ω_∆).

One denotes by eu(K) the restriction of µeu(K) to the zero-section M of T^∗M and calls it the Euler class of K. Hence

euM(K)∈H_Supp(K)⁰ ∩∆(M;ωM).

(5.4)

It is nothing but the class induced by the composition k_∆M −→K −→ω_∆M. We say thatL∈D^b(kM) isinvertibleifLis locally isomorphic tok_M[d] for some d∈Z. Then, L^⊗−1:= RHom(L,k_M) is also invertible andL⊗^LL^⊗−1 ≃ k_M.

Proposition 5.3. Let L be an invertible object in D^b(kM) and K a trace kernel. Then K⊗^L (L⊠^L L^⊗−1) is a trace kernel and µeu K⊗^L (L⊠^L L^⊗−1)

= µeu(K).

Proof. L⊠^L L^⊗−1 is canonically isomorphic to k_M×M on a neighborhood of

the diagonal set ∆_M of M ×M. Q.E.D.