量子形變模上的兩個定理

國立臺灣大學理學院數學系 博士論文

Department of mathematics College of science National Taiwan University

Ph.D Thesis

量子形變模上的兩個定理

Two theorems for deformation quantization modules

陳厚伊 Hou-Yi Chen

指導教授 _: 陳榮凱 博士 advisor : Dr. Jung-Kai Chen

中華民國₉₈年₁₀月 October, 2009

Table of Contents

Introduction . . . 1

1 Preliminary . . . 4

2 Review on the GAGA theorem . . . 12

3 Review on the results of Laumon . . . 17

4 The first main theorem . . . 21

5 Applications to D-modules . . . 30

6 Review on DQ-modules (after K-S) . . . 33

7 Analytization of a DQ-algebroid . . . 42

8 The second main theorem . . . 44

References. . . .53

Acknowledgements

First of all, I would like to thank my advisor Prof. Jungkai Alfred Chen for teaching me algebraic geometry and supporting me when I studied in Taiwan.

Also, I like to express my thanks to Prof. Pierre Schapira for introducing me the theory of deformation quantization modules and helping me with many discussions.

Finally I am grateful to National Taiwan University and the French institute in Taipei for offering me scholarship, without which I would not have been able to complete this research.

Abstract to the Dissertation

Two theorems for deformation quantization modules The theory of deformation quantization modules have a great improvement recently. In this thesis, we prove two basic theo- rems about this theory.

The first theorem is a generalization of Riemann-Roch the- orem for D-modules. We generalize the (algebraic) Riemann- Roch theorem for D-modules of [16] to (analytic) cW -modules.

The second theorem is a generalization of Serre’s GAGA the- orem [see 6]. Let X be a smooth complex projective variety with associated compact complex manifold X_an. If AX is a DQ- algebroid on X, then there is an induced DQ-algebroid on X_an. We show that the natural functor from the derived category of bounded complexes of A^X-modules with coherent cohomologies to the derived category of bounded complexes of A^Xan-modules with coherent cohomologies is an equivalence.

Introduction

The theory of deformation quantizaton modules (we usually call DQ-modules) was introduced and developed by M. Kashi- wara and P. Schapira recently. In this thesis, two basic theorems are proved.

Let X be a smooth complex algebraic variety or a complex manifold. Set C^~ := C[[~]] and C^~,loc := C((~)). In [13], M.

Kashiwara and P. Schapira introduce the notion of a DQ-algebra A^X which is a C^~ := C[[~]]-algebra and locally isomorphic to an algebra (O^X[[~]], ?) where ? is a star product. They also consider the notion of a DQ-algebroid, that is, a C^~-algebroid (in the sense of stacks) locally equivalent to the algebroid associated with a DQ-algebra.

If AX is a DQ-algebra or a DQ-algebroid on X, we denote by AX^a the opposite algebra or algebroid A_X^op and we denote by A^X1×X2 the external product of A^Xi (i = 1, 2).

If A^X is a DQ-algebra or a DQ-algebroid on X, then we have the notion ofA^X-modules. We denote by Mod(A^X) the category of A^X-modules, by D^b(A^X) its bounded derived category and by D^b_coh(A^X) the full triangulated subcategory of the bounded de- rived category D^b(A^X) with coherent cohomologies. An object of D^b(A^X) is called a kernel.

Similarly, set AX^loc := C^~,loc ⊗^L_C~A^X, then we have the notion of good AX^loc-modules. We denote by Mod(AX^loc) the category of AX^loc-modules, by D^b(AX^loc) its bounded derived category and by D^b_gd(AX^loc) the full triangulated subcategory of the bounded derived category D^b(AX^loc) with good cohomologies.

Let (X,A^X), (Y,A^Y) be two complex manifolds endowed with DQ-algebras A^X and A^Y. Let M ∈ D^b(A^X) and K ∈ D^b(A^Xa×Y) be two kernels. Their convolution is defined as

M ◦ K := Rq2!(K ⊗^L_AX q₁⁻¹M )

here, q_i the i-th projection defined on X × Y (i = 1, 2). One of the main theorems in [13] asserts that if M and K are coherent and if q₂ is proper on q⁻¹₁ supp(M ) ∩ supp(K ), then M ◦ K is coherent.

Similarly, IfM1 ∈ D^b_gd(AX^loc) and K1 ∈ D^b_gd(AX^loca×Y) are good modules and if q₂ is proper on q₁⁻¹supp(M¹) ∩ supp(K¹), then their convolution M¹ ◦K¹ is also good.

If X and Y are compact, then a kernel K ∈ D^bcoh(A^Xa×Y) defines a functor

◦K : D^bcoh(A^X) → D^b_coh(A^Y)

which is called the Fourier-Mukai transform induced by K . Similarly, a kernel K ^loc ∈ D^b_gd(AX^loca×Y) defines a functor

◦K ^loc : D^b_gd(AX^loc) → D^b_gd(AY^loc)

which is called the Fourier-Mukai transform induced by K ^loc. The first main theorem of this thesis is the Riemann-Roch theorem for Fourier-Mukai transforms of A^loc-modules.

As an application, we recover the Riemann-Roch formula for D-modules of [16].

If (X, A^X) is a smooth complex algebraic variety endowed with a DQ-algebroid, then there is an induced DQ-algebroid A^Xan on the complex manifold X_an induced by X. Then we con- struct a functor f^∗ : D^b_coh(A^X) → D^b_coh(A^Xan). The second main theorem of this thesis is the following:

Assume that X is projective. Then the functor f^∗: D^b_coh(AX) → D^b_coh(A^Xan) is an equivalence.

This thesis is organized as follows: In Chapter 1, we briefly review the notions of coherent modules, derived categories and the projective limits of 2-categories which are used in the text.

In Chapter 2, we review Serre’s GAGA theorem and translate this theorem to the derived version. In Chapter 3, we review the Riemann-Roch theorem for D-modules in [16]. We prove the first main theorem in Chapter 4. In Chapter 5, we use the main results in Chapter 4 to recover the results in Chapter 3. In Chapter 6, we reveiw the notions and results in [13], in particu- lar, Remark 6.6 and Finiteness theorem 6.11 are crucial to the proof of the second main theorem. We show how to induce an analytic DQ-algebroid from an algebraic DQ-algebroid in Chap- ter 7. In the final chapter, we prove the second main theorem.

Note: Throughout this thesis, all varieties (or schemes) are over C if not otherwise specified.

1 Preliminary

1.1 Coherency

Let A^X be a sheaf of rings on a topological space X.

Definition 1.1.

(1) A (left)AX-module F is said to be locally finitely generated if for each x ∈ X there exist an open neighborhood U of x, an integer N ∈ Z≥0, and epimorphism of A^X|_U-modules on U , (A^X|_U)^⊕N  F |^U.

(2) F is said to be locally finitely presented if for each x ∈ X there exist an open neighborhood U of x, integers N₀, N₁ ∈ Z^≥0, and exact sequence of A^X|_U-modules on U ,

(A^X|_U)^⊕N1 → (A^X|_U)^⊕N0 →F |^U → 0.

(3) F is said to be pseudo-coherent if for every open set U all locally finitely generated A^X|_U-submodules of F |^U are locally finitely presented.

(4) A pseudo-coherent and locally finitely generatedA^X-module is said to be coherent.

We denote by

Mod(A^X): the category of A^X-modules.

Then Mod(AX) is a Grothendieck category [14, Theorem 18.1.6].

Recall that a Grothendieck category C is an abelian category such that C admits a generator and inductive limits and filtrant inductive limits are exact. In particular, Mod(A^X) has enough injectives.

From basic properties of coherent modules (see [11, Proposi- tion A.2 and Proposition A.6]), one can see that the category of coherent AX-modules denoted by

Mod_coh(AX)

is a full abelian thick subcategory of Mod(A^X). Recall that the term thick means that it is closed by kernels, cokernels and ex- tensions.

We also denote by

K(Modcoh(A^X)): the Grothendieck group of Modcoh(A^X).

1.2 Homotopy Categories

Let C := Mod(AX) be an abelian category. A complex M in C consists of a family {(Mⁿ, dⁿ_M)}_n∈Z where Mⁿ is an object of C , and dⁿM is a morphism dⁿ_M : Mⁿ → Mⁿ⁺¹, called a differential of M , satisfying dⁿ⁺¹_M ◦ dⁿ_M = 0. Denote by C(C ) the additive category of complexes in C . A morphism f : M → N in C(C ) consists of a family {fⁿ}_n∈Z of morphisms fⁿ : Mⁿ → Nⁿ (n ∈ Z) satisfying dⁿN ◦ fⁿ = fⁿ⁺¹ ◦ dⁿ_M. For a complex M and an integer k, define a complex M [k] by M [k]ⁿ = M^k+n and dⁿ_{M [k]} = (−1)^kd^k+n_M . For f : M → N , define f [k] : M [k] → N [k] by f [k]ⁿ = f^k+n.

A complex M is said to be bounded below if Mⁿ = 0 (n  0), bounded above if Mⁿ = 0(n  0), and bounded if it is bounded above and below. Denote by C⁺(C ), C⁻(C ), and C^b(C ) the full subcategories of C(C ) consisting of complexes of bounded below, bounded above and bounded, respectively.

To X ∈ C , we associate a complex Xⁿ defined by X if n = 0 and 0 if n 6= 0 and dⁿ_X = 0. We thus consider C as a full subcategory of C(C ).

For complexes M and N in C , define a complex Hom(M,N) of modules by

Hom(M ,N )ⁿ = Q

k

Hom_C(M^k, N^k+n).

For f = (f^k) ∈ Hom(M, N )ⁿ (f^k ∈ Hom(M^k, N^k+n)), we define the differential df ∈ Hom(M, N )ⁿ⁺¹ by letting its k-th com- ponent (df )^k : M^k → N^k+n+1 be the sum M^{k f}

k

−→ N^{k+n d}

k+n

−→N

N^k+n+1 and M^{k (−1)}

n+1d^k_M

−→ M^{k+1 f}

k+1

−→ N^k+n+1.

Then d² = 0 and Hom(M, N ) is a complex of A^X-modules.

The 0-th cocycle

Z⁰(Hom(M, N )) := Ker(Hom(M, N )⁰ → Hom(M, N )^{d 1}) is nothing but Hom_{C(C )}(M, N ). We denote by Ht(M, N ) the 0-th coboundary

B⁰(Hom(M, N )) := Im(Hom(M, N )^{−1 d}→ Hom(M, N )⁰);

its element is called a morphism from M to N homotopic to 0.

By the definition, f = (fⁿ) ∈ Hom_{C(C )}(M, N ) is homotopic to 0 if and only if there exists s = (sⁿ : Mⁿ → Nⁿ⁻¹)_n∈Z such that

fⁿ = dⁿ⁻¹_N sⁿ+ sⁿ⁺¹dⁿ_M. Define a new additive category K(C ) by

Ob(K(C )) := Ob(C(C )),

Hom_{K(C )}(M, N ) := H⁰(Hom(M, N )) = Hom_{C(C )}(M, N )/Ht (M, N ).

The composition of morphisms is the one induced from C(C ).

Then K(C ) is an additive category. Similarly to the case C(C ), we consider C as a full subcategory of K(C ).

1.3 Triangulated Categories

The category C(C ) is also an abelian category, but K(C ) is not. Instead, K(C ) has a structure called a triangulated cate- gory.

For a morphism f : M → N in C(C ), define a new com- plex M (f ), called a mapping cone of f , as follows: M (f )ⁿ = Mⁿ⁺¹ ⊕ Nⁿ, and its differential dⁿ_{M (f )} : M (f )ⁿ → M (f )ⁿ⁺¹ is the map such that Mⁿ⁺¹ −→ M (f )^{n d}

n M (f )

−→ M (f )ⁿ⁺¹ is the sum of M^{n+1 f}

n+1

−→ Nⁿ⁺¹ −→ M (f )ⁿ⁺¹ and M^{n+1 −d}

n+1

−→ MM ⁿ⁺² −→

M (f )ⁿ⁺¹ and Nⁿ −→ M (f )^{n d}

n M (f )

−→ M (f )ⁿ⁺¹ equals N^{n d}

n

−→N

Nⁿ⁺¹ −→ M (f )ⁿ⁺¹. Then dⁿ⁺¹_{M (f )} ◦ dⁿ_{M (f )} = 0 and thus M (f ) is a complex.

Define morphisms α_f : N → M (f ) and β_f : M (f ) → M [1] in C(C ) by

αⁿ_f : Nⁿ → Mⁿ⁺¹ ⊕ Nⁿ = M (f )ⁿ,

β_fⁿ : M (f )ⁿ = Mⁿ⁺¹ ⊕ Nⁿ → Mⁿ⁺¹ = M [1]ⁿ.

A diagram X → Y^f → Z^g → X[1] in K(^h C ) is called a triangle.

A morphism from a triangle X → Y → Z → X[1] to a triangle X⁰ → Y⁰ → Z⁰ → X⁰[1] is a commutative diagram

X −−→ Y −−→ Z −−→ X[1]

ξ



 y

η





y ^ζ





y ^ξ[1]



 y X⁰ −−→ Y⁰ −−→ Z⁰ −−→ X⁰[1].

If ξ, η and ζ are isomorphisms, these two triangles are said to be isomorphic. A triangle X → Y → Z → X[1] in K(C ) is said to be distinguished if there exists a morphism X^{0 f}→ Y⁰ such that the triangle is isomorphic to X^{0 f}→ Y^{0 α}→ M (f )^f → X^{βf 0}[1].

With the automorphism [1] : K(C ) → K(C ) sending M to M [1] and distinguished triangles, one can see that K(C ) is a triangulated category.

Similarly, one can define the distinguished categories K⁺(C ), K⁻(C ) and K^b(C ).

1.4 Derived Categories

A morphism f : X → Y in K(C ) is called a quasi-isomorphism if Hⁿ(X) → Hⁿ(Y ) are isomorphisms for all n.

We obtain the derived category from K(C ) by regarding quasi- isomorphisms are isomorphisms. More precisely, we define the derived category D(C ) as follows:

Define the family Ob(D(C )) of objects in D(C ) to equal Ob(K(C )). For X, Y ∈ Ob(D(C )) = Ob(K(C )), define the family Hom_{D(C )}(X, Y ) of morphisms to be S(X, Y )/ ∼, where S(X, Y ) is a family given in the following, and ∼ is its equiva- lence relation. S(X, Y )/ ∼ denotes the family of ∼ equivalence classes in S(X, Y ).

(1) S(X, Y ) is the set of pairs (s, f ) where s : X⁰ → X is a quasi-isomorphism and f : X⁰ → Y is a morphism in K(C ).

(2) for (s₁, f₁), (s₂, f₂) ∈ S(X, Y ) where s_i : X_i⁰ → X are quasi- isomorphisms and fi : X_i⁰ → Y for i = 1, 2, we define (s1, f1) ∼ (s2, f2) if there exists quasi-isomorphisms ti : X₀⁰ → X_i⁰ for i = 1, 2 and a morphism g : X₀⁰ → Y such that s₁ ◦ t₁ = s₂ ◦ t₂ and f₁ ◦ t₁ = f₂ ◦ t₂ = g.

Similarly, one can define the derived categories D⁺(C ), D⁻(C ) and D^b(C ). Note that D⁺(C ) (resp. D⁻(C ), resp. D^b(C )) is equivalent to the full subcategory of D(C ) consisting of objects

X such that Hⁿ(X) = 0 for n  0 (resp. n  0, resp. |n|  0).

We denote by

D^b(A^X): the bounded derived category of Mod(A^X), D^b_coh(A^X): the full triangulated subcategory of D^b(A^X)

consisting of complexes with cohomology sheaves belonging to Mod_coh(A^X),

K_coh(AX): the Grothendieck group of D^b_coh(AX).

Note that by [2, P.283 Lemma 1.6], we have (1.1) K(Mod_coh(AX)) ' K_coh(AX).

1.5 Projective limits of 2-cat

Recall that a presite X is nothing but a category which we denote by C_X. If S is a prestack on X, then we have the mor- phism u : U₁ → U₂ in C_X, and the functor r_u : S(U₂) → S(U₁) for U₁, U₂ ∈ C_X.

Definition 1.2. Let S be a prestack on X. We denote by lim←−

U ∈CX

S(U ) the category defined as follows.

(a) An object F of lim

U ∈C←−_X

S(U ) is a family {(F_U, ϕ_u)}_{U ∈CX,u∈Mor(CX)} where:

(i) for any U ∈ C_X, F_U is an object of S(U ),

(ii) for any morphism u : U₁ → U₂ in C_X, ϕ_u : r_uF_U2 −→ F^∼ _U1

is an isomorphism such that for any sequence U₁ → U^u ₂ → U^v ₃ of morphisms in C_X, the following diagram commutes (this is a so-called cocycle condition):

r_ur_vF_U3 −−−→ r^ru(ϕv) _uF_U2

C_u,v



 y



 y

ϕu

r_v◦uF_U3 −−→^ϕv◦u F_U1

(b) For two objects F = {(F_U, ϕ_u)} and F⁰ = {(F_U⁰, ϕ⁰_u)} in lim←−

U ∈CX

S(U ), Hom _lim

←−

U ∈CX

S(U )(F, F⁰) is the set of families f = {f_U}_{U ∈CX} such that f_U ∈ Hom_{S(U )}(F_U, F_U⁰ ) and the following diagram commutes for any u : U₁ → U₂

ruFU₂ ϕu

−−→ F_U1

ru(f_U2)



 y



 y

f_U1

ruF_U⁰₂ ^ϕ

0

−−→ Fu _U⁰ 1. Therefore,

Hom _lim

←−

U ∈CX

S(U )(F, F⁰) ' lim←−

U ∈CX

Hom_S|U(F_U, F_U⁰ ).

For any A ∈ C_X^∧ := Fct(C_X^op, Set), we set S(A) = lim←−

(U →A)∈CA

(S|_A)(U ) = lim←−

(U →A)∈CA

S(U ).

Hence, lim←−

U ∈CX

S(U ) = S(pt_X), where pt_X denotes as usual the terminal object of C_X^∧.

We set

S(X) := S(pt_X) = lim

U ∈C←−X

S(U ).

A morphism v : A → A⁰ in C_X^∧ defines a functor

r_v : S(A⁰) = lim

U →A←−⁰

S(U ) → lim

U →A←−

S(U ) = S(A)

and it is easy to check that the conditions of prestack are satis- fied.

Proposition 1.3. ([14, §19]) Let S be a prestack on the small presite X. Then S extends naturally to a prestack on bX, where X denotes the presite associated with the category Cb _X^∧.

2 Review on the GAGA Theorem

Let X be a scheme of finite type and let X_an be the associated complex analytic space. Denote by Mod(O^X) (resp. Mod(O^Xan)) the category of sheaves on X (resp. X_an) and Mod_coh(O^X) (resp.

Mod_coh(O^Xan)) the full subcategory of coherent sheaves. There is a continuous map ϕ : X_an → X of the underlying topological spaces and there is also a natural map of the structure sheaves ϕ⁻¹OX → OXan. To F ∈ Mod(OX), one associates its complex analytic sheaf F^an := O^Xan ⊗_ϕ⁻¹_OX ϕ⁻¹F ∈ Mod(O^Xan). Hence we obtain a functor:

(∗) Υ_X : Mod(OX) → Mod(OXan).

If F is a coherent sheaf, then F^an is also coherent.

The following theorem for a projective scheme is proved in Serre’s famous paper GAGA (cf [19]) which is generalized by Grothendieck for a proper scheme (cf [8, XII]).

Theorem 2.1. Let X be a projective scheme. Then the functor (∗) induces an equivalence of categories

Mod_coh(O^X) −→ Mod^∼ _coh(O^Xan).

Furthemore, for every coherent sheaf F on X, the natural maps Hⁱ(X;F ) → Hⁱ(X_an;F^an)

are isomorphisms, for all i ≥ 0. 

The following lemma is Theorem 2.2.8 in [5].

Lemma 2.2. Let A⁰ and B⁰ be thick subcategories of abelian categories A and B, respectively, and let Φ : A → B be an exact

functor that takes A⁰ to B⁰. Assume furthermore that the fol- lowing properties are satisfied :

1. A and B have enough injectives;

2. Φ is an equivalence of categories when restricted to A⁰ → B⁰;

3. Φ induces a natural isomorphism

Extⁱ_A(F, G) ∼=Extⁱ_B(Φ(F ), Φ(G)) for any F, G ∈ A⁰ and any i.

Then the natural functor eΦ : D^b_A0(A) → D^b_B0(B) induced by Φ is an equivalence of categories.

Proof. We briefly sketch the proof here for reader’s convenience.

(i) We prove that the functor eΦ is fully faithful, i.e. that for any F^•, G^• ∈ D^b_A0(A), eΦ induces an isomorphism

(2.1) Hom_D^b_(A)(F^•, G^•) ∼= Hom_D^b_(B)(eΦ(F^•), eΦ(G^•)).

We’ll use a technique known as d´evissage to prove it. The d´evissage technique is just induction on the number n(E^•) de- fined as

n(E^•) = max{j − i | H^j(E^•) 6= 0, Hⁱ(E^•) 6= 0}.

Hence we shall prove (1.1) by induction on N = n(F^•) + n(G^•).

If N = −∞, then one of F^• or G^• is the zero complex, so there is nothing to prove. If N = 0, then there exist F ∈ A⁰, G ∈ A⁰ such that F^• = F [a] and G^• = G[b] for some a, b ∈ Z. Then

Hom_D^b_(A)(F^•, G^•)= Hom_D^b_(A)(F [a], G[b])=Ext^b−a_A (F, G) and

Hom_D^b_(B)(eΦ(F^•), eΦ(G^•))= Hom_D^b_(B)(eΦ(F [a]), eΦ(G[b]))

= Ext^b−a_B (Φ(F ), Φ(G)).

Hence (2.1) follows from property 3 above.

Assume that eΦ induces an isomorphism

Hom_D^b_(A)(F^•, G^•) ∼= Hom_D^b_(B)(eΦ(F^•), eΦ(G^•))

for all F^•, G^• ∈ D^b_A0(A) with n(F^•) + n(G^•) < N , and let F^•, G^• be objects of D^b_A0(A) with n(F^•) + n(G^•) = N > 0. We may assume that n(G^•) = N > 0 and that Gⁱ = 0 for i < 0, and H⁰(G^•) 6= 0.

Let G^0• be the complex with single non zero object H⁰(G^•) in degree zero. From the morphism G^0• → G^•, there exists a distinguished triangle G^00• → G^0• → G^• → G^00•[1]. By the long exact cohomology sequence, one deduces n(G^00•) < n(G^•); also, from the assumption, n(G^0•) = 0 < n(G^•). From the long exact sequence of Hom’s, the five-lemma and the induction hypothesis we conclude that

Hom_D^b_(A)(F^•, G^•) ∼= Hom_D^b_(B)(eΦ(F^•), eΦ(G^•))

which is what we needed to prove that eΦ is fully faithful. (The case when n(G^•) = 0 but n(F^•) > 0 follows in a similar way.)

(ii) The functor eΦ is essentially surjective: any object G^• of D^b_B0(B) is isomorphic to an object of the form eΦ(F^•) for some F^• ∈ D^b_A0(A). We prove this by induction on n = n(G^•): the case n = −∞ is trivial, and n = 0 follows from property 2.

So assume n > 0, and construct a distinguished triangle G^00• → G^0• → G^• → G^00•[1] where G^0• = H⁰(G^•) 6= 0 and we assume G^• is zero in degrees < 0. Since Φ is an equivalence of categories between A⁰ and B⁰, we can find an F^0• ∈ D^b_A0(A) such that Φ(F^0•) ∼= G^0•. Also, by the induction hypothesis, we can find an F^00• ∈ D^b_A0(A) such that eΦ(F^00•) ∼= G^00•. Since we proved that eΦ is fully faithful, we can find a map F^00• → F^0•

whose image by eΦ is just the side of the distinguished triangle constructed before. Again, we have the distinguished triangle

F^00• → F^0• → F^• → F^00•[1]. Then, since eΦ is a ∂ functor (a functor commutes with the translation functor and sends a distinguished triangle to a distinguished triangle) because Φ is exact, we see that eΦ(F^•) is isomorphic to G^•, as required.  Let X be a scheme of finite type, set A = Mod(OX), B = Mod(O^Xan), A⁰ = Modcoh(O^X) and B⁰ = Modcoh(O^Xan), and de- note by D^b_coh(X) = D^b_A0(A) and D^b_coh(Xan) = D^b_B0(B). Clearly, A⁰ (resp. B⁰) is a full thick subcategory of A (resp. B).

As an application of Lemma 2.2, we have

Corollary 2.3. Let X be a projective scheme, then the functor Υ_X of (∗) induces an equivalence (we keep the same notation)

Υ_X : D^b_coh(X) −→ D^{∼ b}_coh(X_an).

Proof. We shall apply Lemma 2.2. First note that O^Xan,x is a flat OX,x-module for each x ∈ X, hence the functor Υ_X : A → B is an exact functor. Both A and B have enough injectives [see

§1.1]. The functor Υ_X is an equivalence by Theorem 2.1. Hence it is sufficient to check condition 3 of Lemma 2.2. It is enough to prove that

(2.2) RHom(F , G ) ' RHom(F^an,G^an) for F , G ∈ A⁰. Since RHom(F , G ) ' RΓ(X, F^∗ ⊗^L_OX G ) and RHom(F^an,G^an) ' RΓ(X_an, (F^an)^∗ ⊗^L_OXan G^an), where F^∗ = RH omOX(F , O^X) and (F^an)^∗ = RH omOXan (F^an,OXan), we reduce (2.2) to the following isomorphism

(2.3) RΓ(X,F^•) ' RΓ(X_an, (F^•)^an), where F^• ∈ D^b_coh(X).

Assume that (2.3) holds for F^• ∈ D^b_coh(X) of amplitude ≤ N. Now let F^• be of amplitude N+1. Assume for example

H^j(F^•) = 0 for j /∈ [a, a+N + 1] and consider the distinguished triangle

τ^≤a+NF^• −→F^• −→ H^a+N+1(F^•)[−a − N − 1]−→ .⁺¹ Let b = a + N. Since H^b+1(τ^≤bF^•) = H^b+1(H^b+1(F^•)[−b − 1]) = 0, one deduces that H^j(F^•) = 0 for j /∈ [a, b]. Hence by d´evissage, one reduces F^• to a single sheaf F . By Theorem 2.1, we have RΓ(X,F ) ' RΓ(X^an,F^an) and the result follows.

3 Review on the results of Laumon

Let X, Y be smooth algebraic varieties with sheaf of differential operators D^X and D^Y. For a morphism f : X → Y , we have induced morphisms:

(3.1) T^∗X ←− X ×^fd _Y T^∗Y −→ T^{fπ ∗}Y

where T^∗X and T^∗Y denote the cotangent bundles of X and Y , respectively.

If f is proper, there are well defined functors (see [16, section 5]):

(3.2) f_d^! : D^b_coh(OT^∗X) → D^b_coh(OX×YT^∗Y) defined by

f_d^!(−) := f_d^∗(−) ⊗_OX ω_X/Y

where ω_X/Y = f⁻¹(ω_Y^⊗−1) ⊗_f⁻¹_OY ω_X denotes relative canonical bundle of f

and

(3.3) f_π∗ : D^b_coh(O^X×YT^∗Y) → D^b_coh(O^T∗Y).

Hence the functors of (3.2) and (3.3) induce group homomor- phisms (see [16, 6.2]):

f_d^! : K_coh(OT^∗X) → K_coh(OX×YT^∗Y) and

f_π∗ : K_coh(O^X×YT^∗Y) → K_coh(O^T∗Y).

Let (M , F ) be a filtered D^X-module with filtration {F_iM }.

Recall from [10, Definition 2.1.2] that F is a good filtration of M if the graded module gr^FM is coherent over π^∗O^T∗X where gr^FM := L FiM /Fi−1M and π : T^∗X → X is the projection and set

gr^^FM := O^T∗X ⊗_π⁻¹_{π∗OT ∗X} π⁻¹gr^FM .

If M ∈ Modcoh(DX), then there exists globally a good filtra- tion F on M (see [10, Theorem 2.1.3]). We denote by Car(M ) the element of Kcoh(O^T∗X) defined by ^gr^FM . For a proper mor- phism f : X → Y , we also denote by R

f : K_coh(D^X) → K_coh(D^Y) the group homomorphism induced by the direct image functor R

f : D^b_coh(D^X) → D^b_coh(D^Y) given by R

f M = Rf^∗(D^{Y ←X} ⊗^L_DX M ) for M ∈ D^bcoh(D^X).

The following theorem is the main result of [16].

Theorem 3.1. Let f : X → Y be a proper morphism of smooth algebraic varieties. Then the following diagram is commutative

K_coh(D^X)

R

−−→f K_coh(D^Y)

Car



 y



 y^Car K_coh(O^T∗X) ^{fπ ∗f}

!

−−−→ Kd _coh(O^T∗Y).

 Remark 3.2. Combining (algebraic) Grothendieck-Riemann- Roch formula at the level of cotangent bundles with Theorem 3.1, then we have a Riemann-Roch formula for D-modules.

Recall that for a smooth algebraic variety X, we have the corresponding complex manifold X^an and morphism ι : X^an → X. Then we have a canonical morphism

ι⁻¹D^X →D^Xan of sheaves of rings satisfying

D^Xan ' O^Xan ⊗_ι⁻¹_OX ι⁻¹D^X.

Hence we obtain a functor

(•)^an : Mod(D^X) → Mod(D^Xan)

sending M to M^an := D^Xan⊗_ι⁻¹_DX ι⁻¹M . Note that since D^Xan is faithfully flat over ι⁻¹(D^X), this functor is exact. We denote by Mod_gd(D^Xan) the category of good D^Xan-modules (for defini- tion and its properties, see [11, 4.7]), in particular, Mod_gd(D^Xan) is an abelian thick subcategory of Mod(D^Xan). We also de- note by D^b_gd(D^Xan) the full triangulated subcategory of D^b(D^Xan) consisting of complexes with cohomology sheaves belonging to Mod_gd(DX^an) and K_gd(DX^an) the Grothendieck group of D^b_gd(DX^an).

Recall from [10, Corollary 1.4.17] that for a coherent D^X- module M , M is generated by a coherent O^X-module F , i.e.

M = D^X ·F . Denote by

FiD^X := {P ∈ E nd_C(O^X) | [P, f ] ∈ Fl−1D^X, ∀f ∈ O^X} and Mⁱ := FiD^X ·F ,

then Mi ⊂ Mi+1 for each i ≥ 0 and M = S

i

Mi = lim−→

i

Mi. Hence we have the well defined functor:

Υ : Mod_coh(DX)−→Mod_gd(DX^an) sending M = S

i

Mi to M^an := S

i

Mi^an. On the other hand, if X is proper, then by GAGA theorem, we have the well defined inverse funtor to the functor Υ. Hence we get the equivalence:

Modcoh(D^X) −→ Mod^∼ _gd(D^Xan).

In particular, by [2, P.283 Lemma 1.6], we have (3.4) K_coh(DX) ' K_gd(DX^an).

For a morphism f : X → Y of smooth algebraic varieties, we have the induced analytic morphism of complex manifolds f^an : X^an → Y^an and the corresponding morphisms to (3.1):

T^∗X^{an f}

an

←− Xd ^an×_Y^an T^∗Y^{an f}

an

−→ Tπ ^∗Y^an.

If f is a proper morphism, then we have the corresponding functors to (3.2) and (3.3):

(3.5) f_d^an! : D^b_coh(O^T∗Xan) → D^b_coh(O^Xan×Y anT^∗Y^an) defined by

f_d^an!(−) := f_d^an∗(−) ⊗_OXan ω_X^an_/Y^an and

(3.6) f_π^an_∗ : D^b_coh(OX^an×Y anT^∗Y^an) → D^b_coh(OT^∗Y^an).

Hence (3.5) and (3.6) induce group homomorphisms:

f_d^an! : K_coh(OT^∗X^an) → K_coh(OX^an×Y anT^∗Y^an) and

f_π^an_∗ : K_coh(OX^an×Y anT^∗Y^an) → K_coh(OT^∗Y^an).

Hence, by Theorem 3.1 and (3.4), we get following theorem.

Theorem 3.3. Let f : X → Y be a proper morphism of proper smooth algebraic varieties, then we have the following commu- tative diagram

K_gd(DX^an)

R

−−→f an K_gd(DY^an)

Car



 y



 y^Car K_coh(O^T∗Xan) ^f

an π ∗f_d^an!

−−−−→ K_coh(O^T∗Yan) where the group homomorphism R

f^an is induced by the direct im- age functor R

f . 

Remark 3.4. As in Remark 3.2. By using (analytic) Riemann- Roch formula for O-modules, we get a Riemann-Roch formula for analytic D-modules.

4 The first main theorem

4.1 Star-products

Let X be a complex manifold (or a smooth variety). We denote by δ_X : X ,→ X × X the diagonal embedding and we set 4_X = δ_X(X). We denote by OX the structure sheaf on X, by Ω_X the sheaf of differential forms of maximal degree and by Θ_X the sheaf of vector fields. As usual, we denote by DX the sheaf of rings of differential operators on X. Recall that a bi-differential operator P on X is a C-bilinear morphism O^X×O^X → O^X which is obtained as the composition δ_X⁻¹ ◦ ˜P where ˜P is a differential operator on X × X defined on a neighborhood of the diagonal and δ⁻¹ is the restriction to the diagonal:

P (f, g)(x) = ( ˜P (x₁, x₂; ∂_x1, ∂_x2)(f (x₁)g(x₂))|_x1=x2=x. Hence the sheaf of bi-differential operators is isomorphic to

D^X ⊗_OX D^X,

where the both D^X are regarded as O^X-modules by the left mul- tiplications.

Definition 4.1. A star algebra on O^X[[~]] is a C^~-bilinear sheaf morphism

? : OX[[~]] × OX[[~]] → OX[[~]]

satisfies the following conditions:

(i) the star product makes OX[[~]] into a sheaf of associated unital C^~-algebra with unit 1 ∈ OX.

(ii) there is a sequence P_i : O^X × O^X → O^X of bi-differential operators, such that for any two local sections f, g ∈ OX

one has

f ? g = f g +

∞

P

i=1

P_i(f, g)~ⁱ.

Note that f ? g ≡ f g mod ~, and Pi(f, 1) = P_i(1, f ) = 0 for all f and i > 0. We call (O^X[[~]], ?) a star algebra.

4.2 DQ-algebras

Definition 4.2. A DQ-algebra A on X is a C^~-algebra locally isomorphic to a star-algebra (O^X[[~]], ?) as a C^~-algebra.

Clearly, a DQ-algebra is a sheaf of ~-adically complete flat C^~-algebra on X satisfying A /~A ' OX. Note also that for an algebraic variety X, a DQ-algebra A is called deformation quantization of O^X in [3] and [20].

Remark 4.3. For a smooth projective variety X, there exists a DQ-algebra A^X on X. For details, one refers to [3].

4.3 Riemann-Roch theorem for DQ-modules

Let X be a complex manifold. To a DQ-algebra AX on X, we associate its ~-localization, the C^~,loc-algebra

AX^loc = C^~,loc⊗_C~A^X. Hence we have an exact functor

Mod(A^X) ^⊗C~C

~,loc

−→ Mod(AX^loc).

IfM is an AX^loc-module, M⁰ is an A^X-submodule andM⁰⊗_C~

C^~,loc −→^∼ M , we shall say that M⁰ generates M .

A coherent AX^loc-module M is good if for any open relatively compact subset U of X, there exists a coherent AX|_U-module which generates M |U.

We denote by Mod_gd(AX^loc) the thick abelian subcategory of Mod (AX^loc) consisting of good AX^loc-modules. We denote by D^b_gd(AX^loc) the full triangulated subcategory of D^b(AX^loc) consist- ing of objects with cohomology sheaves belonging to Mod_gd(AX^loc).

We also denote by K(Mod_gd(AX^loc)) the Grothendieck group of Modgd(AX^loc) and Kgd(AX^loc) the Grothendieck group of D^b_gd(AX^loc) and note that

(4.1) K(Mod_gd(AX^loc)) ' K_gd(AX^loc) by [2, P. 283 Lemma 1.6].

If (Y,A^Y) is another complex manifold endowed with a DQ- algebra A^Y, denote by q_i the i-th projection defined on X × Y (i = 1, 2). Let M ∈ D^bcoh(AX) and K ∈ D^bcoh(AX^a×Y). Set

M ◦ K := Rq^2!(K ⊗^L_AX q₁⁻¹M ).

If q₂ is proper on q⁻¹₁ supp(M )∩supp(K ), then by [13, Theorem 9.1], M ◦ K ∈ D^bcoh(A^Y).

Similarly, let M¹ ∈ D^b_gd(AX^loc) and K¹ ∈ D^b_gd(AX^loca×Y). Set M1 ◦K1 := Rq_2!(K1

⊗L_A^loc

X q⁻¹₁ M1).

If q₂ is proper on q₁⁻¹supp(M1) ∩ supp(K1), then M1 ◦ K1 ∈ D^b_gd(AY^loc) (see [18, Corollary 3.3.5]).

If X and Y are compact, then a kernel K ∈ D^bcoh(A^Xa×Y) defines a functor

(4.2) ◦K : D^bcoh(A^X) → D^b_coh(A^Y)

which is called the Fourier-Mukai transform induced by K . Hence the Fourier-Mukai transform of (4.2) defines a group ho- momorphism of Grothendieck groups

◦[K ] : K^coh(A^X) → Kcoh(A^Y)

where [K ] ∈ K^coh(A^Xa×Y). Similarly, a kernel K ^loc belongs to D^b_gd(AX^loca×Y) defines a functor

(4.3) ◦K ^loc : D^b_gd(AX^loc) → D^b_gd(AY^loc)

which is called the Fourier-Mukai transofrm induced by K ^loc and the Fourier-Mukai transform of (4.3) defines a group homo- morphism of Grothendieck groups

◦[K ^loc] : K_gd(AX^loc) → K_gd(AY^loc) where [K ^loc] ∈ K_gd(AX^loca×Y).

Denote by

(4.4) gr : D^b_coh(A^X) → D^b_coh(O^X) the functor M → C⊗^L_C~ M (see [13]).

We have the following proposition.

Proposition 4.4. Let (X,AX) and (Y,AY) be two compact complex manifolds endowed with DQ-algebras AX and AY and let K ∈ D^bcoh(AX^a×Y). Then the following diagram is commu- tative

(4.5)

D^b_coh(AX) −−→ D^{◦K b}_coh(AY)

gr



 y



 y

gr

D^b_coh(OX) −−−−→ D^{◦(grK ) b}_coh(OY) which induces the following commutative diagram

K_coh(A^X) −^◦[−−^{K ]}→ K_coh(A^Y)

gr



 y



 y

gr

K_coh(O^X) −−−−→ K^{◦[grK ]} _coh(O^Y)

where the group homomorphisms gr (4.4) are induced by (4.5).

Proof. Using the fact that the functor gr commutes with the

convolution ◦. 

Definition 4.5. Let X be a compact complex manifold, and denote by H^∗(X, C) = L

i

Hⁱ(X, C). One defines the Mukai vector of an object E ∈ D^b_coh(O^X) as the cohomology class

υ : K_coh(O^X) → H^∗(X, C) by the formula

υ([E]) = ch([E]).ptd(X)

where ch([E]) is the Chern character of [E] and td(X) is the Todd class of tangent bundle of X.

Let X and Y be compact complex manifolds and let E ∈ D^b_coh (O^X×Y). Define the cohomological integral transform associated to E

Φ_[E] : H^∗(X, C) → H^∗(Y, C)

by Φ_[E](α) = q_2∗(υ([E].q₁^∗(α)), where q_i denotes the i-th projec- tion defined on X × Y (i = 1, 2).

We have the following theorem.

Theorem 4.6.([5, Proposition 3.1.9] or [9, Corollary 5.29]) The following diagram is commutative

K_coh(O^X) −−→ K^◦[E] _coh(O^Y)

υ



 y



 y^υ H^∗(X, C) −−→ H^{Φ[E] ∗}(Y, C).

 Combining Proposition 4.4 with Theorem 4.6, we obtain the following theorem.

Theorem 4.7.(Riemann-Roch for A -modules) Let (X, A^X) and (Y,A^Y) be two compact complex manifolds endowed with DQ- algebras A^X and A^Y and let K ∈ D^bcoh(A^Xa×Y). Then the fol- lowing diagram is commutative

K_coh(AX) −^◦[−−^{K ]}→ K_coh(AY)

υ◦gr



 y



 y

υ◦gr

H^∗(X, C) −−−→ H^{Φ[grK ] ∗}(Y, C).

 4.2 Riemann-Roch theorem for A^loc-modules

First, we need the following lemma.

Lemma 4.8. Let (X,AX) be a compact complex manifold en- dowed with a DQ-algebra AX. Let M ∈ Modgd(AX^loc) and let M⁰ ⊂ M which generates M . Then [M⁰/~M⁰] belongs to K(Modcoh(O^X)) depends only on M .

Proof. We consider another generator M0⁰ of M . Since X is compact, there exists m, n ≥ 0 such that M0⁰ ⊂ ~⁻ⁿM⁰ and M⁰ ⊂ ~^−mM0⁰. Hence

M0⁰ ⊂ ~⁻ⁿM⁰ ⊂ ~^−m−nM0⁰. Since our modules have no ~-torsion, we have

~⁻ⁿM⁰/~⁻ⁿ⁺¹M⁰ ' M⁰/~M0.

Hence we may assume that for N large enough, we have