THE LOCAL DUALITY THEOREM IN D -MODULE THEORY by

THE LOCAL DUALITY THEOREM IN D -MODULE THEORY by

Luis Narv´ aez Macarro

Abstract. — These notes are devoted to the Local Duality Theorem forD-modules, which asserts that the topological Grothendieck-Verdier duality exchanges the de Rham complex and the solution complex of holonomic modules over a complex analytic manifold. We give Mebkhout’s original proof and the relationship with Kashiwara-Kawai’s proof. In that way we are able to precise the commutativity of some diagrams appearing in the last one.

Résumé (Le théorème de dualité locale dans la théorie desD-modules). — Ce cours est consacr´e au th´eor`eme de du alit´e locale pour lesD-modules, qui affirme que la dualit´e topologique de Grothendieck-Verdier ´echange le complexe de de Rham et le complexe des solutions des modules holonomes sur une vari´et´e analytique complexe. On donne la preuve originale de Mebkhout en faisant le rapport avec la preuve de Kashiwara- Kawai. Ceci nous permet de pr´eciser la commutativit´e de certains diagrammes dans cette derni`ere.

Contents

Introduction . . . 60

Notation . . . 61

1. Duality for Analytic Constructible Sheaves . . . 61

2. The Local Duality Morphism inD-module Theory . . . 64

3. Proof of the Local Duality Theorem . . . 67

Appendix . . . 79

References . . . 86

2000 Mathematics Subject Classification. — 32C38, 32S60.

Key words and phrases. — de Rham complex, Grothendieck-Verdier duality.

Supported by DGES PR97, BFM2001-3207 and FEDER.

Introduction

These notes are issued from a course taught in the C.I.M.P.A. School on Differential Systems, held at Seville (Spain) from September 2 through September 13, 1996. They are an improved version of the handwritten notes distributed during the School.

The aim of these notes is to introduce the reader to the Local Duality Theorem in D-module Theory —LDT for short— and to explain in a detailed way the proofs of it in [Me3], [K-K]. This theorem asserts that the Verdier duality for analytic cons- tructible complexes interchanges the “De Rham” and the “Solutions” of every bounded holonomic complex ofD-modules on a complex manifold. Besides the importance and the beauty of such a result, it is a good representative of the relationship between discrete and continuous coefficients, an important idea in contemporary Algebraic Geometry.

The first published duality type result is a punctual one due to Kashiwara [Ka],

§5. The LDT in the way we currently use was first stated by Mebkhout in [Me1], 4.1, [Me2], 5.2, but its proof depended on a still conjectural theory of Topological Homological Algebra. A complete proof was given in [Me3], III.1.1 (see also [Me4], 1.1, [Me5], ch. I, 4.3). Kashiwara and Kawai proposed another proof in [K-K], 1.4.6 based on the punctual result above.

The proof of the punctual result of Kashiwara uses the Local Duality in Analytic Geometry (residues). Mebkhout’s proof of the LDT uses Serre and Poincar´e-Verdier dualities to construct the duality morphism and to prove it is an isomorphism. Kashi- wara and Kawai define the duality morphism as the formal one and reduce the proof of the LDT to the former result of Kashiwara by means of the Biduality Theorem for analytic constructible complexes. However, this reduction demands the commutati- vity of some diagram involving the global formal duality morphism and the punctual one, which is not obvious. Both proofs are evidently based on the Kashiwara’s Con- structibility Theorem.

In these notes we prove that the duality morphism defined by Mebkhout coincides with the formal one and, as a consequence, that the diagram needed in Kashiwara- Kawai’s proof is commutative. This fact is explained by the relationship between the Global Serre Duality and the Local Duality in Analytic Geometry (cf.[Li]).

As we could expect, to do the task we need to be especially attentive to the de- finition and the properties of the different formal objects involved. In particular, we have to manage some signs. A complete reference for these questions is [De2], 1.1.

For the sake of completeness and for the ease of the reader, we have collected (a big portion of) them in the Appendix.

Other somewhat different proofs of the LDT are available in [Bo2], §19, [Sa], 2.7, [Bj], III, 3.3.10. We have chosen to present the first proof of the LDT, due to Mebkhout, and the proof of Kashiwara-Kawai because they are conceptually simple and they fit in this collective work as a continuation of [M-S1, M-S2].

This work has been done during a sabbatical year at the Institute for Advanced Study, Princeton. I would like to thank this institution for its hospitality. Discussions with Pierre Deligne have been of great value to me. I am grateful to him. I am also grateful to Leo Alonso and Ana Jerem´ıas for good suggestions.

Notation

Given a sheaf of rings RX on a topological space X, we shall denote by C^∗(RX), K^∗(RX) and D^∗(RX) the category of complexes, the homotopy category of complexes and the derived category of the abelian category of left RX-modules respectively. We shall use _rRX for referying to the category of rightRX-modules.

The symbolsA^•,B^•,C^•,etc. will be used for complexes of sheaves on a topological space: the objects of A^• are theAⁿ and the differentials are dⁿ_A : Aⁿ → Aⁿ⁺¹, for every n∈Z.

Given a complexA^•and an integerd, we shall denote byh^d(A^•) itsdth cohomology object.

Given a complexA^• (of objects in some additive category), the complexA^•[1] is defined byA^•[1]ⁿ =Aⁿ⁺¹,d_A•[1]=−d_A.

The total derived functors of Hom^•_RX(−,−),Hom^•_RX(−,−) and −⊗^•_RX − will be denoted by RHom^•_RX(−,−),RHom^•_RX(−,−) and of −⊗⊗⊗^•_RX − respectively, and Ext^d_R

X(−,−) =h^dRHom^•_R

X(−,−).

IfRXis the constant sheaf associated to a fixed ringKand no confusion is possible, we shall abreviate Hom^•_KX(−,−),Hom^•_KX(−,−),RHom^•_KX(−,−) andExt^d_KX(−,−) by Hom^•_X(−,−),Hom^•_X(−,−),RHom^•_X(−,−) andExt^d_X(−,−) respectively.

1. Duality for Analytic Constructible Sheaves

Throughout this sectionX denotes a connected complex analytic manifold count- able at infinity of dimension d, and D_c^b(CX) the derived category of bounded com- plexes of sheaves ofC-vector spaces with analytic constructible cohomology (cf.[Ve], [Ka], [M-N3]). We denoteTX=CX[2d].

1.1. The Topological Biduality Morphism. — The abelian category of sheaves of complex vector spaces overX has finite injective dimension (cf.[DP], exp. 2, 4.3).

The functorRHom^•_X(−,−) induces a functor

RHom^•_X(−,−) :D^b(CX)×D^b(CX)−→D^b(CX)

which can be computed by taking injective resolutions of the second argument, or locally free resolutions of the first argument if they exist.

1.1.1. Proposition. — If F^•1,F^•2 are two complexes in D_c^b(CX), then RHom^•_X(F1^•,F2^•) is also inD^b_c(CX). Furthermore, if theF^•i are constructible with respect to a Whitney stratification Σof X, thenRHom^•_X(F^•1,F^•2)is also constructible with respect to Σ.

Proof⁽¹⁾. — We can suppose that the F^•i are single constructible sheaves Fi

(cf.[M-N3], II.5). The question being local (cf. loc. cit., I.4.21) we can sup- pose that F1 = σ!L, for σ : S ,→ X the inclusion of a stratum of Σ and L a local system (of finite rank) on S (cf. loc. cit., I.4.14). In this case we have RHom^•_X(σ_!L,F2) ' Rσ_∗RHom^•_S(L, σ^!F2), and we can conclude by induction on the dimension ofX and Thom-Whitney’s isotopy theorem (cf. loc. cit., I.4.15).

1.1.2. Definition. — For every bounded complexF^• inD_c^b(CX) we define itsdual by F^•∨:=RHom^•_X(F^•,CX)

and thetopological biduality morphism β_F• :F^•→(F^•∨)^∨ as in A.2.

1.1.3. Proposition. — If F^• is a bounded constructible complex on X, then for each point x ∈ X and for every small ball B centered in x with respect to some local coordinates, the complex RΓ_c(B,F^•)has finite dimensional cohomology.

Proof. — According to proposition 1.1.1, the complex F^•∨ is bounded and cons- tructible. Then, for every small ball B centered in x, the canonical morphism RΓ(B,F^•∨) → (F^•∨)_x is an isomorphism (cf.[M-N3], I.4.16) and we conclude by the Poincar´e-Verdier duality

RΓ(B,F^•∨) =RHom^•_B(F^•|_B,CB) '

−−−→RHom^•_C(RΓ_c(B,F^•),C)[−2d]

(cf.[DP], exp. 5).

1.2. The Biduality Theorem. — The Biduality Theorem for analytic con- structible sheaves has been first stated and proved by Verdier in [Ve], 6.2 using Resolution of Singularities. Other proofs in the setting of cohomologically con- structible sheaves are available in [DP], exp. 10, §2, [Bo1], V, 8.10, [K-S], 3.4. We sketch here a proof following the lines in [SGA 4¹₂], Th. finitude, 4.3 and [M-N3], III.2.1,III.2.6 and based on the Poincar´e-Verdier duality cf.[DP], exp. 4,5, [Bo1], V, 7.17, [Iv], VII.5.2, [K-S], 3.1.10.

1.2.1. Theorem. — For each bounded constructible complex F^• on X, the biduality morphism β_F• :F^• →(F^•∨)^∨ is an isomorphism.

Proof. — We can suppose thatF^•is a single constructible sheafF(cf.[M-N3], II.5).

The result is clear ifFis a local system (of finite rank).

(1)This proof is also valid in the case of an arbitrary complex analytic space.

As the question is local, we can also suppose thatX =D₁^d−1×D2, where theDi

are open disks inC,Fis a local system on the complement of an hypersurfaceZ ⊂X and the first projectionp:X →D₁^d−1 is finite overZ (cf. loc. cit., I.4.20).

We can extend our data, first to a constructible sheaf Fe on Xe =D^d−1₁ ×C and second to F = σ!eF, where σ : X ,e → X = D₁^d−1×P1 is the (open) inclusion. Call p:X →Y =D₁^d−1 the first projection, which is proper.

Let us consider the triangle

(1) F β_F

−−−→(F^∨)^∨−→Q^•−→F[1]

where the support of the (bounded) complexQ^• is contained inZ∪(Y × {∞}) and then it is finite overY.

By taking direct images bypwe obtain a new triangle inD_c^b(CY) Rp_∗F Rp_∗β_F

−−−−−−−→Rp_∗(F^∨)^∨−→Rp_∗Q^•−→Rp_∗F[1]

(cf.[M-N3], I.4.23).

In order to prove that β_F is an isomorphism we need to prove that Q^• = 0, but that is equivalent toRp_∗Q^• = 0 becausepis finite over the support ofQ^•.

Let TrX/Y :Rp_∗TX →TY be the topological trace morphism for the proper map p. According to the local form of the Poincar´e-Verdier duality (cf.[Iv], VII.5, [K-S], 3.1.10) the morphism ρ_K• composition of

Rp_∗RHom^•_X(K^•,TX) nat.

−−−−→RHom^•_Y(Rp_∗K^•,Rp_∗TX) (Tr_X/Y)_∗

−−−−−−−−→RHom^•_Y(Rp_∗K^•,TY) is an isomorphism for every bounded complex of sheaves ofC-vector spacesK^•.

Callρ^∗

F :=RHom^•_Y(ρ_F,TY) the isomorphism induced byρ_F. According to A.5, we can “redefine”

(F^∨)^∨=RHom^•_X(RHom^•_X(F,T_X),T_X) and using A.2 and lemma A.15 we deduce the relation

ρ_RHom• X(F,T_X)

◦Rp_∗β_F=ρ^∗

F◦β_Rp

∗F. By induction hypothesis, the morphism β_Rp

∗F is an isomorphism, then Rp_∗β_F too and we obtain the desiredRp_∗Q^•= 0.

1.2.2. As X is a connected oriented manifold of (topological) dimension 2d, the topological trace morphism tr_X : H^2d_c (X,CX)→ C given by integration of topC^∞- forms with compact support is an isomorphism. Then, for each pointx∈X, denoting by i : {x} ,→ X the inclusion, the canonical morphism i^!CX → RΓc(X,CX) gives rise to apunctual topological traceisomorphism

trx: H^2d_x (CX) nat.

−−−−→

' H^2d_c (X,CX) tr_X

−−−−→

' C.

1.2.3. Proposition. — LetF^• be a complex inD^b_c(CX)andx∈X. Denotei:{x},→X the (closed) inclusion. Then, the natural morphism

n: (F^•)^∨_x =i⁻¹RHom^•_X(F^•,CX)−→RHom^•_C(i^!F^•, i^!CX) is an isomorphism. In particular, using 1.2.2, we obtain an isomorphism

((F^•)^∨)x'RHom^•_C(i^!F^•,C)[−2d].

Proof. — As (F^•)^∨_x is a bounded complex ofC-vector spaces with finite dimensional cohomology andi^!CX'C[−2d], the natural morphism A.2

β₀: (F^•)^∨_x −→RHom^•_C(RHom^•_C((F^•)^∨_x, i^!CX), i^!CX) is an isomorphism. We also have a canonical isomorphism (cf.A.11)

g:i^!((F^•)^∨)^∨=i^!RHom^•_X((F^•)^∨,CX) '

−−−→RHom^•_C((F^•)^∨_x, i^!CX).

Call g^∗ := RHom^•_C(RHom^•_C(g, i^!CX), i^!CX) the isomorphism induced by g, and (i^!β_F•)^∗:=RHom^•_C(i^!β_F•, i^!CX) the morphism induced byi^!β_F•, which is an isomor- phism according to theorem 1.2.1. To conclude, we observe thatn= (i^!β_F•)^∗◦g^∗◦β₀ according to A.12.

2. The Local Duality Morphism in D-module Theory

Throughout this section X denotes a complex analytic manifold countable at in- finity of dimension d, DX the sheaf of linear differential operators with coefficients inOX (cf.[G-M], I) and D^b_c(DX) the derived category of bounded complexes of left DX-modules with coherent cohomology.

2.1. The Solution and the De Rham Functors. — Here, our basic functor is RHom^•_D

X(−,−) which can be computed by taking injective resolutions of the second argument, or locally free resolutions of the first argument if they exist.

SinceDX is a coherent sheaf of rings and every singleDX-module admits locally a finite free resolution (cf.[Me5], ch. I, 2.1.16), we have an induced functor

RHom^•_DX(−,−) :D_c^b(DX)×D^b(DX)−→D^b(CX).

TheDe Rham functor is DR=RHom^•_D

X(OX,−) :D^b(DX)−→D^b(CX) and theSolutions functors are

S=Hom^•_DX(−,OX) :K^b(DX)−→K^b(CX), S=RHom^•_DX(−,OX) :D_c^b(DX)−→D^b(CX).

We will also consider the“external” duality functors D=Hom^•_D

X(−,DX) :K_c^b(DX)−→K_c^b(_rDX), D=RHom^•_D

X(−,DX) :D^b_c(DX)−→D_c^b(_rDX).

We haveS(DX) =S(DX) =OX.

The De Rham functor can be computed by means of the Spencer resolution Sp^•_X (cf.[Me5], ch. I, 2.1.17), whose objects are defined by Sp^−p_X = DX⊗_OX ∧^p Der_C(OX), p= 0, . . . , dand the differential^−p:Sp^−p_X →Sp^−(p−1)_X is given by:

^−p(P⊗(δ1∧ · · · ∧δp)) =

p

X

i=1

(−1)ⁱ⁻¹(P δi)⊗(δ1∧ · · · ∧δbi∧ · · · ∧δp)

+ X

16i<j6p

(−1)^i+jP⊗([δ_i, δ_j]∧δ₁∧ · · · ∧δb_i∧ · · · ∧δb_j∧ · · · ∧δ_p)

forp= 2, . . . , dand⁻¹(P⊗δ) =P δforp= 1.

There is an obvious augmentation⁰:Sp⁰_X=DX →OX,⁰(P) =P(1), that makes Sp^•_Xinto a (canonical) locally free resolution ofOXas leftDX-module. We will always consider this augmentation to identify the functors DR(−) =Hom^•_DX(Sp^•_X,−).

Every left DX-moduleE carries an integrable connection ∇ : E → Ω¹_X⊗_OX E and we can then consider itsclassical De Rham complex Ω^•_X(E) (cf.[De1], I.2). It is defined by Ω^p_X(E) = Ω^p_X⊗_OX E forp= 0, . . . , d, and the differential∇^p : Ω^p_X(E)→ Ω^p+1_X (E) is given by∇^p(ω⊗e) = (dω)⊗e+ (−1)^pω∧ ∇(e).

2.1.1. Lemma. — For each leftDX-moduleE, the morphisms α^p_E: Ω^p_X⊗_OXE−→Hom^p_D

X(Sp^•_X,E) =Hom_DX(Sp^−p_X ,E), p= 0, . . . , d defined byα^p_E(θ⊗e)(P⊗δ) = (−1)^p(p+1)/2P· hδ, θi ·e, commute with the differentials and gives rise to a natural isomorphism of complexes

α^•_E: Ω^•_X(E)−→Hom^•_DX(Sp^•_X,E).

The proof of the lemma is straightforward. It should be noticed that the sign (−1)^p(p+1)/2is imposed by the definition of the functorHom^•_D

X(−,−) (cf.A.1).

We will denote

α^•₀:=α_D^•_X : Ω^•_X(DX) '

−−−→Hom^•_DX(Sp^•_X,DX) =D(Sp^•_X) =D(OX), α^•₁:=α^•_OX : Ω^•_X = Ω^•_X(OX) '

−−−→Hom^•_DX(Sp^•_X,OX) =S(Sp^•_X) =S(OX).

Obviouslyα^•₀is right DX-linear.

2.1.2. Denote by ωX the sheaf of top differential forms Ω^d_X on X. It carries a canonical right DX-module structure (cf.[G-M], prop. 15, [M-N2], 1.1.5). Call σ^• : Ω^•_X(DX)→ωX[−d] the right DX-linear morphism given byσ^d(θ⊗P) =θ·P. It is a quasi-isomorphism (cf.[Me5], ch. I, 2.6.6) admitting a CX-linear section τ^• given byτ^d(θ) =θ⊗1. Let us consider the following morphisms:

α^• :=α^•₀◦τ^• :ωX[−d]−→Hom^•_DX(Sp^•_X,DX) =D(Sp^•_X), α:ωX[−d] α^•₀◦(σ^•)⁻¹

−−−−−−−−−→Hom^•_DX(Sp^•_X,DX) =D(Sp^•_X) can

−−−→

' D(Sp^•_X) =D(OX).

The first one is aC^X-linear quasi-isomorphism, and the second one is an isomorphism in the derived category of rightDX-modules. Both morphisms coincide inD^b(CX).

In particular, the cohomology of the complexDR(DX) vanishes in degree different fromdand thenDR(DX) =Ext^d_D

X(OX,DX)[−d].

According to the Poincar´e lemma, the inclusion CX ⊂Ω⁰_X gives rise to a quasi- isomorphism κ^•₀:CX →Ω^•_X = Ω^•_X(OX). Using the isomorphism of complexesα^•₁ we obtain an isomorphism in the derived category

(2) κ:CX

−−−'→Hom^•_DX(Sp^•_X,OX) =S(OX) =DR(OX).

2.2. The Duality Morphism

2.2.1. Definition. — For every bounded complex of leftDX-modulesM^•with coherent cohomology we define the duality morphism

ξ_M• :DR(M^•)−→S(M^•)^∨=RHom^•_CX(S(M^•),CX) by composing the natural morphism (cf.A.2)

ξ:RHom^•_DX(OX,M^•)−→RHom^•_CX(S(M^•),S(OX)) with the isomorphism induced byκ(2).

2.2.2. Proposition. — ForM^• ∈D_c^b(DX)there exist (local) natural isomorphisms λ_M^• :DR(DX)⊗⊗⊗^•_DX M^• −→DR(M^•),

µ_M• :S(DX)^∨⊗⊗⊗^•_DX M^• −→S(M^•)^∨ such that the following diagram commutes

DR(DX)

•

⊗⊗⊗_DXM^• ξ_DX ⊗Id_M^•

//

λ_M^• '

S(DX)^∨

•

⊗⊗

⊗_DX M^• ' µ_M•

DR(M^•) ξ_M•

//S(M^•)^∨.

Proof. — Take a flat resolution P^• → M^• and an injective Godement resolution OX→J^•. We have

DR(DX)⊗⊗⊗^•_DX M^• =Hom^•_D

X(Sp^•_X,DX)⊗^•_DXP^•,

DR(M^•) =Hom^•_DX(Sp^•_X,M^•) =Hom^•_DX(Sp^•_X,P^•), S(DX)^∨⊗⊗⊗^•_DX M^• =O^∨X

•

⊗⊗⊗_DX M^• =Hom^•_C

X(J^•,Hom^•_D

X(Sp^•_X,J^•))⊗^•_DX P^•, S(M^•)^∨=· · ·=Hom^•_CX(Hom^•_DX(P^•,J^•),Hom^•_DX(Sp^•_X,J^•)) and we are reduced to lemma A.10.

The fact that λ_M^• and µ_M• are isomorphisms is a local question. So, we can suppose that M^• has a finite free resolution and we are reduced to the obvious fact thatλ_DX andµ_DX are isomorphisms.

2.2.3. Corollary. — For every bounded complex of left DX-modules M^• with coherent cohomology, the duality morphism ξ_M• is an isomorphism if and only if ξ_D

X ⊗Id_M^• is an isomorphism.

3. Proof of the Local Duality Theorem

Throughout this section X denotes a complex analytic manifold countable at in- finity of dimensiond.

3.1. Statement of the Local Duality Theorem

3.1.1. Theorem. — For every bounded complex of leftDX-modulesM^• with holonomic cohomology, the duality morphism

ξ_M• :DR(M^•)−→S(M^•)^∨ is an isomorphism (in the derived category).

3.2. The Basic Commutative Diagram

3.2.1. Proposition ([Me3], [Me4], [Me5]). — The complex S(DX)^∨ = O^∨X is concen- trated in degreed= dimX.

Proof. — For every integeri>0, the sheafExtⁱ_CX(OX,CX) is the sheaf associated to the presheafU 7→Extⁱ_CU(OU,CU).It is enough to prove that Extⁱ_CU(OU,CU) = 0 for alli6=dand for every Stein open setU ⊂X.

Now, by the Poincar´e-Verdier duality (cf.[DP], exp. 5, [Iv], VI) the space Extⁱ_CU(OU,CU) is isomorphic to the algebraic dual of H^2d−i_c (U,OU), and by the Serre duality [Se], ifUis Stein, the space H^2d−i_c (U,OU) is isomorphic to the topological dual of H^i−d(U, ωU), but for such open sets H^i−d(U, ωU) = 0 and then Extⁱ_CU(OU,CU) = 0, for alli6=d.

3.2.2. Call

ξ:=h^d(ξ_DX) :Ext^d_DX(OX,DX)−→Ext^d_CX(OX,CX), α:=h^d(α) :ω_X −→Ext^d_D

X(OX,DX)

whereα is the isomorphism in 2.1.2. Both morphisms are rightDX-linear.

AsO^∨X is concentrated in degreed, for every open setU ⊂X we have Γ(U,Ext^d_CX(OX,C^X)) =R^dΓ(U,Ext^d_CX(OX,C^X)[−d])

=R^dΓ(U,RHom^•_CX(OX,CX)) =h^dRHom^•_CU(OU,CU), and using the natural isomorphismν^d (cf.A.5) we obtain an isomorphism

εU : Γ(U,Ext^d_CX(OX,C^X)) '

−−−→Hom_D(CU)(OU,C^U[d]).

The εU commute with restrictions and each εU is right DX(U)-linear, where the right DX(U)-module structure on Hom_D(CU)(OU,CU[d]) comes from the left action ofDX(U) onOU.

The Poincar´e quasi-isomorphism κ^•₀ : CX → Ω^•_X and the inclusion map κ^•₁ : ω_X[−d]→Ω^•_X gives rise to aPoincar´e-De Rham morphism in the derived category

κ⁰:= (κ^•₀[d])⁻¹◦κ^•₁[d] :ωX−→CX[d].

We will denote byβ_U : Γ(U, ω_X)→Hom_D(CU)(OU,CU[d]) the composition of (κ⁰)_∗ with the map

Γ(U, ωX) = Hom_OU(OU, ωU) forget

−−−−−−→Hom_CU(OU, ωU) = Hom_D(CU)(OU, ωU).

In corollary 3.2.5 we will see thatβ_U is rightDX(U)-linear.

Recall that (cf.2.1.2)

α^•=α^•₀◦τ^•:ωX[−d]−→Hom^•_DX(Sp^•_X,DX) =D(Sp^•_X), and denote

β^•:= (κ^•₁)∗ ◦(forget) :ωX[−d]−→Hom^•_CX(OX,Ω^•_X), where “(forget)” is the morphism

ωX[−d] =Hom^•_OX(OX, ωX[−d]) forget

−−−−−−→Hom^•_CX(OX, ωX[−d]), and

γ^•:= (α^•₁)_∗:Hom^•_CX(OX,Ω^•_X) '

−−−→Hom^•_CX(OX,S(Sp^•_X)).

3.2.3. Proposition. — The following diagram of complexes of sheaves of vector spaces

(3)

D(Sp^•_X) ξ^• //Hom^•_CX(OX,S(Sp^•_X))

ωX[−d] β^• //

α^•

OO

Hom^•_C

X(OX,Ω^•_X) γ^• '

OO

is commutative.

Proof. — Asαⁱ =βⁱ = 0 for all i6=d, we need only to prove thatξ^d◦α^d=γ^d◦β^d, butS(DX) =OX is a complex vanishing in degrees different from 0 and then there is no signs in the expression for ξ^d (cf.A.2). We deduce that the degreed part of the diagram (3) can be identified with the diagramm

(4)

Hom_DX(Sp^−d_X ,DX) nat.//Hom_CX(OX,Hom_DX(Sp^−d_X ,OX))

ω_X forget //

α^d

OO

Hom_CX(OX, ωX).

(α^d₁)_∗ '

OO

For a top differential formθon an open setU ⊂X, the section

ϕ=α^d(θ)∈Γ(U,Hom_DX(Sp^−d_X ,DX)) = Hom_DU(Sp^−d_U ,DU) is given by

ϕ(P⊗δ) =α^d₀(θ⊗1)(P⊗δ) = (−1)^d(d+1)/2P· hδ, θi.

Call ψ ∈ Γ(U,Hom_CX(OX,Hom_DX(Sp^−d_X ,OX))) = Hom_CU(OU,Hom_DU(Sp^−d_U ,OU)) the morphism determined byϕ. For each local sectionf ofOU we have

ψ(f)(P⊗δ) =ϕ(P⊗δ)(f) = (−1)^d(d+1)/2(P· hδ, θi)(f) = (−1)^d(d+1)/2P(hδ, θif).

On the other hand, the section

ϕ⁰ = forget(θ)∈Γ(U,Hom_CX(OX, ωX)) = Hom_CU(OU, ωU) is given byϕ⁰(f) =f θ. Call

ψ⁰ = (α^d₁)∗(ϕ⁰) =α^d₁◦ϕ⁰∈Hom_CU(OU,Hom_DU(Sp^−d_U ,OU)).

We have

ψ⁰(f)(P⊗δ) =α^d₁(ϕ⁰(f))(P⊗δ) =α^d₁(f θ)(P⊗δ) = (−1)^d(d+1)/2P(hδ, f θi) and we conclude that the diagram (4) is commutative, and then (3) too.

3.2.4. Proposition. — For every open set U ⊂ X, the following diagram of vector spaces

Γ(U,Ext^d_D

X(OX,DX)) Γ(U, ξ) //Γ(U,Ext^d_C

X(OX,CX)) εU

' Γ(U, ω_X) β_U //

Γ(U, α) '

OO

Hom_D(CU)(OU,CU[d]) is commutative.

Proof. — Let us call

• a_U :h^dΓ(U, ω_X[−d])−→⁼ Γ(U, ω_X) the identity map,

• bU = the composition of h^dΓ(U,D(Sp^•_X)) can

−−−−→R^dΓ(U,D(Sp^•_X)) (can)

−−−−−→R^dΓ(U,D(Sp^•_X)) =R^dΓ(U,D(OX))

=R^dΓ(U,Ext^d_DX(OX,DX)[−d]) = Γ(U,Ext^d_DX(OX,DX)),

• cU = the composition of

h^dΓ(U,Hom^•_CX(OX,S(Sp^•_X))) can

−−−−→R^dΓ(U,Hom^•_CX(OX,S(Sp^•_X))) (can)

−−−−−→R^dΓ(U,RHom^•_CX(OX,S(Sp^•_X))) =R^dΓ(U,RHom^•_CX(OX,S(OX))) (κ⁻¹)_∗

−−−−−→

' R^dΓ(U,O^∨X) =R^dΓ(U,Ext^d_C

X(OX,C^X)[−d]) = Γ(U,Ext^d_C

X(OX,C^X)),

• anddU = the composition of h^dΓ(U,Hom^•_C

X(OX,Ω^•_X)) can

−−−−→R^dΓ(U,Hom^•_C

X(OX,Ω^•_X)) (can)

−−−−−→R^dΓ(U,RHom^•_CX(OX,Ω^•_X)) =h^dRHom^•_CU(OU,Ω^•_U) ν^d

−−→' Hom_D(CU)(OU,Ω^•_U[d]) (κ^•₀)⁻¹_∗

−−−−−→

' Hom_D(CU)(OU,CU[d]).

We are going to prove that the following relations:

Γ(U, α)◦aU =bU◦h^dΓ(U, α^•), Γ(U, ξ)◦bU =cU◦h^dΓ(U, ξ^•), εU◦cU◦h^dΓ(U,(γ^•)) =dU, βU◦aU =dU◦h^dΓ(U, β^•) hold, and then we can conclude by using proposition 3.2.3.

The first relation Γ(U, α)◦a_U =b_U◦h^dΓ(U, α^•) is an straightforward consequence of the facts that α^• and α induce the same isomorphism in D^b(C^X) (cf.2.1.2), and that the isomorphisms α and α[−d] coincide after the canonical identification D(OX) =Ext^d_DX(OX,DX)[−d].

The second relation Γ(U, ξ)◦bU =cU◦h^dΓ(U, ξ^•) comes from the standard proper- ties of the total derived (bi)functorsRHom^•(−,−) and of the natural morphism

ξ:RHom^•_D

X(−,?)−→RHom^•_C

X(S(?),S(−))

(cf.A.4), and from the fact that the morphisms ξ_DX and ξ[−d] coincide after the canonical identificationsD(OX) =Ext^d_DX(OX,DX)[−d],O^∨X=Ext^d_CX(OX,CX)[−d].

The third relation ε_U◦c_U◦h^dΓ(U,(γ^•)) = d_U follows from the commutativity of the following diagram

R^dΓ(U,RHom^•_C

X(OX,C^X) (κ^•₀)∗ //

=

R^dΓ(U,RHom^•_C

X(OX,Ω^•_X))

=

h^dRHom^•_CU(OU,CU) (κ^•₀)_∗ //

ν^d

h^dRHom^•_CU(OU,Ω^•_U) ν^d

Hom_D(CU)(OU,C^U[d]) (κ^•₀[d])_∗ //Hom_D(CU)(OU,Ω^•_U[d]) and from standard naturality properties.

The last relationβU◦aU =dU◦h^dΓ(U, β^•) is a consequence of the commutativity of the following diagramms (see A.7)

h^dHom^•_C

U(OU,Ω^•_U) can //

ν₂^d

h^dRHom^•_C

U(OU,Ω^•_U) ν^d

Hom_K(CU)(OU,Ω^•_U[d]) can //Hom_D(CU)(OU,Ω^•_U[d]) and

h^dΓ(U, ωX[−d]) h^dΓ(U, β^•) //

=

h^dΓ(U,Hom^•_CX(OX,Ω^•_X))

=

Γ(U, ωX) β⁰ //

forget

h^dHom^•_CU(OU,Ω^•_U) ν₂^d

Hom_K(CU)(OU, ωU) (κ^•₁[d])_∗ //Hom_K(CU)(OU,Ω^•_U[d])

where β⁰(θ) is the cohomology class of Γ(U, β^d)(θ) for every top differential formθ onU.

3.2.5. Corollary. — For every open set U ⊂X, the morphism βU : Γ(U, ωX)−→HomD(CU)(OU,CU[d]) is right DX(U)-linear.

3.3. Compatibility of the Duality Morphism with the Serre and the Poincar´e-Verdier Dualities: Mebkhout’s Proof

3.3.1. For each open set U ⊂ X we consider the analytic trace morphism Tr_U : H^d_c(U, ωU)→ Cand the topological trace morphism trU : H^2d_c (U,CU) →Cgiven by integration of top differential forms (of type (d, d)) with compact support:

The smooth De Rham complex

0−→CX−→E⁰X −→ · · · −→E^2dX −→0

gives rise to a morphism in the derived categoryθ₁ :E^2dX[−2d]→CX which induces another one

θ₁: Γ_c(U,E^2dX) =R^2dΓ_c(U,E^2dX[−2d]) R^2dΓc(U, θ1)

−−−−−−−−−−−→H^2d_c (U,CU).

The topological trace morphism is defined by the relation: trU◦θ1=R

U. In a similar way, the Dolbeault resolution

0−→ωX−→E^d,0X

−−→ · · ·∂ ∂

−−→E^d,dX =E^2dX −→0

gives rise to a morphism in the derived categoryθ₂:E^2dX[−d]→ω_X inducing another one

θ₂: Γ_c(U,E^2dX) =R^dΓ_c(U,E^2dX[−d]) R^dΓ_c(U, θ₂)

−−−−−−−−−−−→H^d_c(U, ω_U).

The analytic trace morphism is defined by the relation: Tr_U◦θ₂=R

U. The Poincar´e-De Rham morphismκ⁰:ω_X→CX[d] induces a map

β_U⁰ : H^d_c(U, ωU)−→H^d_c(U,CU[d]) = H^2d_c (U,CU).

A straightforward computation shows thatκ⁰◦θ2= (−1)^dθ1[d], and then we obtain

(5) tr_U◦β_U⁰ = (−1)^dTr_U.

Theanalytic Serre pairing [Se]

h−,−i_S : Γ(U, ω_X)×H^d_c(U,OU)−→C

is given by the composition of the analytic trace morphism TrU with the Yoneda pairing

Γ(U, ωX)×H^d_c(U,OU) forget×Id

−−−−−−−−−−→Hom_CU(OU, ωU)×H^d_c(U,OU) Yoneda

−−−−−−−→H^d_c(U, ωU).

The vector space Γ(U, ω_X) has a natural Fr´echet-Schwartz structure and, ifU is Stein, the pairingh−,−iS identifies H^d_c(U,OU) with the topological dual Γ(U, ωX)⁰. Then, H^d_c(U,OU) carries a natural DFS structure and Γ(U, ωX)'H^d_c(U,OU)⁰(cf.[Se], [B-S], ch. 1,§1 (c), 2.1).

ThePoincar´e-Verdier pairing (cf.[DP], exp. 5)

h−,−iP V : HomD(CU)(OU,CU[d])×H^d_c(U,OU)−→C

is given by the composition of the topological trace morphism trU with the Yoneda map

Hom_D(CU)(OU,CU[d])×H^d_c(U,OU) Yoneda

−−−−−−−→H^2d_c (U,CU).

According to the Poincar´e-Verdier duality, the pairing h−,−i_{P V} indentifies Hom_D(CU)(OU,CU[d]) with the algebraic dual H^d_c(U,OU)^∗ (cf. loc. cit.).

3.3.2. Lemma. — The Serre pairing isDX(U)-balanced.

Proof. — Let c be a class in H^d_c(U,OU), θ ∈ Γ(U, ωX) and P ∈ DX(U). For each i, j = 0, . . . , d let E^i,jU be the sheaf of smooth differential forms of type (i, j). The Dolbeault resolution

0−→OU −→E^0,0U

−−→ · · ·∂ ∂

−−→E^0,dU −→0 (resp. 0−→ω_U −→E^d,0U

−−→ · · ·∂ ∂

−−→E^d,dU −→0) is a complex of left (resp. right)DU-modules, and the morphism

θ∧:E^0,•U −→E^d,•U

is a lifting ofθ:OU →ωU. Letα∈Γc(U,E^0,dU ) a section representing the classc. We have

hθ, P ci_S =· · ·= Z

U

θ∧(P α), hθP, ci_S =· · ·= Z

U

(θP)∧α.

Both integrals coincide when P is a holomorphic function. For the general case we can work in local coordinates z = (z1, . . . , zd), z = (z1, . . . , zd), and it is enough to consider P = ∂z_i. Let α = f dz, θ = gdz be the local expressions. We have P α = f_zidz, θP = P^t(g)dz = −g_zidz, where P^t is the transposed operator. The differencehP c, θiS− hc, θPiS is the integral of the closed form (f g)z_idzdz, and then it vanishes.

3.3.3. Lemma. — The Poincar´e-Verdier pairing is DX(U)-balanced.

Proof. — This is a consequence of the easy fact that the Yoneda map Hom_D(CU)(OU,CU[d])×H^d_c(U,OU) (−,−)

−−−−−−→H^2d_c (U,CU)

isDX(U)-balanced. To see that, takec∈H^d_c(U,OU),ϕ∈Hom_D(CU)(OU,CU[d]) and P ∈DX(U)⊂Hom_CU(OU,OU). Then we have

(ϕ, P c) = (ϕ, P_∗(c)) =ϕ_∗(P_∗(c)) = (ϕP)_∗(c) = (ϕP, c).

3.3.4. Proposition. — The following relation

h−,−i_{P V}◦(β_U×Id) = (−1)^dh−,−i_S holds.

Proof. — According to the definition ofβU, the following diagram Γ(U, ω_X)×H^d_c(U,OU) Yoneda //

β_U ×Id

H^d_c(U, ω_U) β_U⁰

Hom_D(CU)(OU,CU[d])×H^d_c(U,OU) Yoneda //H^2d_c (U,CU) is commutative. The propostion then follows from (5).

3.3.5. Proposition. — For each Stein open setU ⊂X, there exist natural rightDX(U)- linear isomorphisms

H^d_c(U,OU)⁰−→^' Γ(U,Ext^d_D

X(OX,DX)) H^d_c(U,OU)^∗−→^' Γ(U,Ext^d_C

X(OX,CX)) such that the following diagram

Γ(U,Ext^d_D

X(OX,DX)) Γ(U, ξ) //Γ(U,Ext^d_C

X(OX,CX))

H^d_c(U,OU)⁰ inclusion //

'

OO

H^d_c(U,OU)^∗ '

OO

commutes.

Proof. — It is a consequence of propositions 3.2.4, 3.3.4, of lemmas 3.3.2, 3.3.3 and of Serre and Poincar´e-Verdier dualities.

3.3.6. According to 2.1.2, corollary 2.2.3 and proposition 3.2.1, the question in the theorem 3.1.1 is equivalent to prove that

ξ⊗Id_M^• :Ext^d_DX(OX,DX)⊗⊗⊗^•_DX M^•−→Ext^d_CX(OX,C^X)⊗⊗⊗^•_DX M^• is an isomorphism.

We can suppose (cf.[M-N3], II.5) that M^• is a single holonomic moduleM. The problem being local, we can also suppose that there exists a finite free resolutionP^•

0−→D^rX^m −→ · · · −→D^rX⁰ −→M−→0.

We have to prove that

ξ⊗Id_P^• :Ext^d_DX(OX,DX)⊗_DX P^•−→Ext^d_CX(OX,CX)⊗_DXP^• is a quasi-isomorphism.