Elliptic Pairs I. Relative Finiteness and Duality

Pierre Schapira Jean-Pierre Schneiders

Contents

1 Introduction 6

2 Elliptic pairs and regularity 8 2.1 Relative -modules...... 8 D 2.2 Relative f-characteristicvariety...... 12 2.3 Relative elliptic pairs ...... 16

3Tools 17 3.1Dolbeaultcomplexeswithparameters...... 18 3.2Realiﬁcationwithparameters...... 19 3.3TrivializationofIR-constructiblesheaves...... 22

3.4 Topological S-modules...... 24 O 4 Finiteness 26 4.1Thecaseofaprojection...... 26 4.2Thegeneralcase...... 27

5 Duality 29 5.1Therelativedualitymorphism...... 30 5.2Thecaseofaclosedembedding...... 36 5.3Thecaseofaprojection...... 38 5.4Thegeneralcase...... 39

6 Base change and K¨unneth formula 40 6.1Basechange...... 40 6.2 K¨unneth formula ...... 46

7 Microlocalization 47

7.1 The topology of the sheaf Y X(0)...... 47 C | 7.2 Direct image and microlocalization ...... 50

8 Main corollaries 53 8.1Extensiontothenonpropercase...... 53 8.2Specialcasesandexamples...... 55

5 1 Introduction

Let f : X Y be a morphism of complex analytic manifolds, a coherent module −→ M over the ring X of differential operators on X, F an IR-constructible object on X.In D this first paper, we give a criterion insuring that the derived direct images of the X - D module F are coherent Y -modules, and we prove related duality and Künneth ⊗M D formulas. Part of these results were announced in [20, 21]. In [22], making full use of these results, we shall associate to ( ,F ) a characteristic M class and show its compatibility with direct image, thus obtaining an index theorem generalizing (in some sense) the Atiyah-Singer index theorem as well as its relative version [1, 3]. Let us describe our results with more details, beginning with the non-relative case for the sake of simplicity.

An elliptic pair on a complex analytic manifold X is the data of a coherent X - D module and an IR-constructible sheaf F on X (more precisely, objects of the derived M categories), these data satisfying the transversality condition

char( ) SS(F) T ∗ X. (1.1) M ∩ ⊂ X Here char( ) denotes the characteristic variety of , SS(F) the micro-support of F M M (see [12]) and TX∗ X the zero section of the cotangent bundle T ∗X. This notion unifies many classical situations. For example, if is a coherent X - M D module, then the pair ( ,CI X ) is elliptic. If U is an open subset of X with smooth M boundary ∂U, the pair ( ,CI U ) is elliptic if and only if ∂U is non characteristic for . M M If X is the complexification of a real analytic manifold M,then( ,CI M ) is an elliptic M pair if and only if is elliptic on M in the classical sense. If F is IR-constructible M on X,then( X,F) is an elliptic pair. If is a coherent X-module, we can associate O G O to it the coherent X -module X, and the results obtained for the elliptic pair D G⊗OX D ( X,CI X ) will give similar results for .See 8 for a more detailed discussion. G⊗OX D G § If f : X Y is a morphism of complex analytic manifolds, we generalize the −→ preceding definition and introduce the notion of an f-elliptic pair, replacing in (1.1) char( )bycharf( ), the f-characteristic variety of (this set was already defined M M M in [19] when f is smooth). The main results of this paper assert that if the pair ( ,F )isf-elliptic, f is proper M on supp( ) supp(F )and is endowed with a good filtration, then: M ∩ M

1) the direct image (in the sense of -modules) f ( F)has Y-coherent coho- D ! M⊗ D mology,

2) the duality morphism

f (D0F D ) D f ( F) ! ⊗ XM −→ Y ! M⊗

is an isomorphism (here, D denotes the dualizing functor for -modules and D0 D is the simple dual for sheaves),

6 3) there is a K¨unneth formula for elliptic pairs,

4) direct image commutes with microlocalization.

See Theorem 4.2, Theorem 5.15, Theorem 6.7 and Theorem 7.5 below for more details. In fact, we obtain these results in a relative situation over a smooth complex manifold S, working with the rings of relative differential operators. This relative setting makes notations a little heavy but it gives us the freedom on the base manifold we need in the proofs. Even if we want the final result over a base manifold reduced to a point, in the proofs, we need to use other bases. So, it is better to work in a relative situation everywhere. Moreover, the base change Theorem 6.5 is a natural way to get the Künneth formula for elliptic pairs. The idea of the proof of the finiteness result goes as follows. First, using the graph embedding, we are reduced to prove the theorem for a closed embedding (this one does not offer much difficulty) and for a projection. Then, using the same trick as in [8], we reduce to the case Y = S. Then it remains to treat the case where X = Z S, f : X S is the second projection, is a X S-module endowed × −→ M D | with a good filtration and F = G CI S where G is an IR-constructible sheaf on Z.We × call it the projection case and we have to prove that in this case Rf!(F X) X S ⊗M⊗D | O is S coherent and S dual to Rf!R om (F ,ΩX S[dX dS]). X S O O H D | ⊗M | − For that purpose, we “trivialize” F by replacing it by a bounded complex of sheaves of the form α CI U ,theUα’s being relatively compact subanalytic open subsets of X ⊕ α satisfying the regularity condition:

D (CI )=CI . 0 Uα Uα

This construction is made possible thanks to the triangulation theorem and a result of Kashiwara [11].

Next, we consider the relative realification IR S of obtained by adding the rel- M | M ative Cauchy-Riemann system to the Z S S -module and remark that since is D × | M M assumed to be good we may always find a resolution of IR S by finite free ZIR S S - M | D × | modules near subsets of ZIR S of the form K ∆whereKis a compact subset × × of Z and ∆ is an open polydisc in S. Moreover, since the solutions of the relative Cauchy-Riemann system are the same in the sheaves of analytic functions, differentiable functions or distributions with holomorphic parameters in S, we can compute the holomorphic solutions of as the relative analytic, differentiable or distributional M solutions of IR S . M | Now, the elliptic hypothesis insures the regularity theorem, that is, the isomorphism

L L F X R om(D0F, X). X S ∼ X S ⊗M⊗D | O −→ H M⊗D | O

Applying Rf! to this isomorphism, we shall compute both sides using the trivialization of F and a ﬁnite free resolution of the relative realiﬁcation of using analytic (resp. M

7 diﬀerentiable) solutions for the left (resp. right) hand side. This will give us a continuous

S-linear quasi-isomorphism O · · (1.2) R1 −→∼ R 2 where the components of the left (resp. right) hand side are DFN-free (resp. FN-free) topological modules over the Fréchet algebra S. The coherence then follows from an O extension of Houzel’s finiteness theorem [7] due to one of the authors [25]. Note that we found no way of applying the original Houzel’s theorem in our situation since it is not obvious to find the requested chain of nuclear quasi-isomorphisms for a given elliptic pair. The duality result is proved along the same lines once we have a clear construction of the general duality morphism which makes it easy to check its compatibility with the various simplifications and transformations used in the proof. Note that the hypothesis that the -module is endowed with a good filtration D M could be relaxed by using cohomological descent techniques as in [24]. However, doing so would have cluttered the proof with unessential technical difficulties. This is why we have preferred to stay to a simpler setting, sufficient for all known applications. Our theorems provide a wide generalization of many classical results as shown in the last section. In particular, we obtain Grauert’s theorem [6] (in the smooth case) on direct images of coherent -modules and the corresponding duality result of Ramis-Ruget-Verdier [15, O 16]. Since we treat -modules, we are allowed to “realify” the manifolds by adding D the Cauchy-Riemann system to the module, and the rigidity of the complex situation disappears, which makes the proofs much simpler and, may be, more natural than the classical ones. We also obtain Kashiwara’s theorem [9] on direct images of coherent -modules as D well as its extension to the non-proper case of [8] (whose detailed proof had never been published) and the corresponding duality result of [23, 24]. In the absolute case, we regain and generalize many well-known theorems concerning regularity, finiteness or duality for -modules (in particular those of [2, 13, 14]), see 8 D § for a more detailed discussion.

2 Elliptic pairs and regularity

2.1 Relative -modules D In this section, we recall some basic facts about relative -modules. D In the sequel, by an analytic manifold we mean a complex analytic manifold X of

ﬁnite dimension dX . Keeping the notations of [12], we denote by

τ : TX X and π : T ∗X X −→ −→ the tangent and cotangent bundles of X.

8 To every complex analytic map f : X Y , we associate the natural maps −→

TX X Y TY TY −→f 0 × −→f τ T∗X X Y T ∗ Y T ∗ Y. t ←−f 0 × −→f π Let S be an analytic manifold. A relative analytic manifold over S is an analytic manifold X endowed with a surjective analytic submersion X : X S .Weoftenuse −→ the notation X S for such an object when we want to avoid confusion on the basis and | set for short dX S = dX dS . | − A morphism f : X S Y S of relative analytic manifolds is the data of a complex | −→ | analytic map f : X Y such that Y f = X . −→ ◦ Let X S be a relative analytic manifold over S. | Since : X S is smooth, the map −→

TX X S TS −→ 0 × is surjective. Its kernel is thus a sub-bundle of TX. WedenoteitbyTX S and call | it the relative tangent bundle of X S. Its holomorphic sections form the sheaf ΘX S of | | vertical holomorphic vector ﬁelds on X S. Recall that a holomorphic vector ﬁeld θ is | vertical if and only if

θ(h X)=0 ◦ for any section h of S. The dual map O

X S T ∗S T ∗ X t × −→ 0 is injective. Its cokernel is thus a quotient-bundle of T ∗X which is isomorphic to the dual of TX S. This is the relative cotangent bundle of X S,wedenoteitbyT∗XS | | | and denote by

pX S : T ∗X T ∗ X S | −→ | p p the canonical projection. The holomorphic sections of T ∗X S form the sheaf ΩX S | | of relative holomorphic diﬀerential forms of degree p. ToV shorten the notations, we set

dX S | ΩX S =ΩXS . | | To every morphism f : X S Y S , we associate the natural maps | −→ |

TX S X Y TY S TY S | −→f 0 × | −→f τ | T∗X S X Y T ∗ Y S T ∗ Y S. t | ←−f 0 × | −→f π | Note that we use the same notations as in the non-relative case since the context will avoid any confusion.

The subring of om ( X , X) generated by the derivatives along vertical holo- H CI X O O morphic vector ﬁelds and multiplication with holomorphic functions is denoted by X S . D | We call it the ring of relative diﬀerential operators on X S. |

9 The basic algebraic properties of X S are easily obtained using the usual ﬁl- D | tration/graduation techniques. We will not review them here and refer the reader to [18, 19].

As usual, we denote by Mod( X S) the abelian category of left X S-modules and D | D | by Coh( X S) the full subcategory of coherent modules. The category Coh( X S)is D | D | a thick subcategory of Mod( X S ) (i.e. it is full and stable by kernel, cokernel and D | extensions).

A coherent X S-module is good if, in a neighborhood of any compact subset of D | M X, admits a finite filtration by coherent X S -submodules k (k =1,...,`)such M D | M that each quotient k/ k 1 canbeendowedwithagoodfiltration.Wedenoteby M M − Good( X S) the full subcategory of Coh( X S) consisting of good X S -modules. This D | D | D | definition ensures that Good( X S) is the smallest thick subcategory of Mod( X S) D | D | containing the modules which can be endowed with good filtrations on a neighborhood of any compact subset of X. b We denote by D( X S ) the derived category of Mod( X S)andbyD( XS)its D | D | D | full triangulated subcategory consisting of objects with bounded amplitude. The full b triangulated subcategory of D ( X S) consisting of objects with coherent (resp. good) D | b b cohomology modules is denoted by Dcoh( X S) (resp. Dgood( X S)). D | D | op We introduce similar notations with the ring X S replaced by the opposite ring X S D | op D | to deal with right X S-modules. Since the categories Mod( X S) and Mod( X S)are D | D | D | equivalent, we will work only in the most convenient one depending on the problem at hand. In the sequel, we will often need to work with bimodule structures. Let k be a field. Recall that if A and B are k-algebras, giving a left (A,B)-bimodule structure on an abelian group M is just giving M a left structure of A-module and a left structure of B-module such that

a (b m)=b(am) · · · · (ca)(bm)=(cb)(am) · · · · · · for any a A, b B, c k and m M. Hence, it is equivalent to consider that ∈ ∈ ∈ ∈ M is endowed with a structure of A B-module. Using this point of view it is easy ⊗k to extend to bimodules the notions and notations deﬁned usually for modules. For example, we will denote by Mod( X S X S) the category of left X S bimodules and D | ⊗D | D | by D( X S X S) the corresponding derived category. D | ⊗D | Let f : X S Y S be a morphism of relative analytic manifolds over S. | −→ | Recall that 1 X S Y S = X f 1O f − Y S D | → | O ⊗ − Y D | has a natural structure of left X S-module compatible with its structure of right 1 D | f − Y S -module. Using this transfer module, we may deﬁne the relative proper di- D | b op rect image of an object of D ( X S) by the formula M D | L f ( )=Rf!( X S Y S) S! X S | M M⊗D | D | → |

10 b op It is an object of D ( Y S). D | Recall also that if (resp. ) is a right (resp. left) X S-module then there is on M N D | a unique structure of right X S-module such that M⊗OX N D | (m n) θ = m θ n m θ n ⊗ · · ⊗ − ⊗ · (m n) h = m h n = m h n ⊗ · · ⊗ ⊗ · for any sections m, n, θ and h of , ,ΘXS and X respectively. M N | O Inthesameway,if , are two left X S-modules then there is on a N P D | N⊗OX P unique structure of left X S-module such that D | θ (n p)=θn p+n θp · ⊗ · ⊗ ⊗· h(n p)=hn p=n hp · ⊗ · ⊗ ⊗ · for any sections n, p, θ and h of , ,ΘXS and X respectively. N P | O Finally, recall the following exchange lemma which will be useful in the sequel.

Lemma 2.1 If is a right X S-module and , are left X S-modules then the M D | N P D | map ( ) ( ) X S X X X S M⊗D | N⊗O P −→ M⊗O N ⊗D | P m (n p) (m n) p ⊗ ⊗ 7→ ⊗ ⊗ is a canonical isomorphism. Let X be a relative analytic manifold over S. Recall that the characteristic variety of a coherent X S -module is a conic analytic subset of T ∗X S denoted by charX S( ) D | M | | M and that 1 char( X )=p− char ( ). X S XS X S D ⊗D | M | | M Hence theorem 11.3.3 of [12] gives the equality

1 SS(R om ( , X)) = p− charX S( ). X S X S H D | M O | | M The sheaf ΩX S of relative holomorphic differential forms of maximal degree is canon- | ically endowed with a structure of right X S-module which is compatible with its struc- D | ture of X-module and characterized by the fact that, for every open subset U of X, O one has ω.θ = Lθω if ω ΩX S(U)andθis a vertical vector field defined on U. − ∈ |

11 As in algebraic geometry, the terminology used in the preceding deﬁnition is justiﬁed by the following biduality result.

Lemma 2.3 There is a canonical sheaf involution of ΩX S X S interchanging its | ⊗OX D | two right X S-module structures. D |

Proof: Letusconsiderthesheaf XS XS where the tensor product uses the left D | ⊗OX D | X module structures of the two copies of X S. This sheaf is obviously endowed with O D | one structure of left X S -module and two structures of right X S-module which are D | D | compatible with each other. The involution

X S X S X S X S D | ⊗OX D | −→ D | ⊗O X D | P Q Q P ⊗ 7→ ⊗ exchanges the two right structures and preserves the left one.

Tensoring over X S with ΩX S using its right structure and the left structure of D | | X S X S and applying the exchange lemma 2.1 gives us the requested involution. D | ⊗OX D | 2

b op Proposition 2.4 For any object of Dcoh( X S), the canonical arrow M D |

R om (R om ( , X S), X S) X S X S M−→ H D | H D | M K | K | deduced from the involution of the preceding lemma is an isomorphism.

Proof: Since is locally isomorphic to a bounded complex of ﬁnite free right X S- M D | modules it is suﬃcient to prove the result for = X S where it is an easy consequence M D | of the preceding lemma and the fact that ΩX S is a locally free X module of rank one. | O 2

It follows that the characteristic variety does not change by duality:

b op Proposition 2.5 If is an object of Dcoh( X S) then one has M D |

charX S( )=charXS(DXS ). | M | | M 2.2 Relative f-characteristic variety In this subsection, we consider a morphism f : X S Y S of relative analytic | −→ | manifolds over S and define the relative characteristic variety charf ( )ofacoherent op M X S-module . First, we consider the case of a relative submersion where such a D | M variety was already defined in [19] for S = pt . Next, by using the graph embed- { } ding, we extend this definition to the general case. Finally, we show how the relative L characteristic variety controls the micro-support of X S Y S. X S M⊗D | D | → |

12 Let f : X S Y S be a relative analytic submersion over S.Sincefis smooth, | −→ | we have the following exact sequence of vector bundles on X:

0 X Y T ∗ Y S T ∗ X S T ∗ X Y 0 . −→ × | −→ψ f | −→φ f | −→ Working as in paragraph III.1.3 of [19] we get the following lemmas.

Lemma 2.6 Assume 0 is a coherent X Y -module. Then M D | 1 char ( 0)=φ− char ( 0). X S X S X Y f X Y | D | ⊗D | M | M

Lemma 2.7 Assume is a coherent X S-module and assume 0, 0 are two co- M D | M N herent X Y -submodules of which generates it as a X S-module then D | M D |

charX Y ( 0)=charXY( 0). | M | N Hence, we may introduce the following deﬁnition.

Deﬁnition 2.8 Let be a coherent X S -module. One deﬁnes the relative charac- M D | teristic variety charf S( ) of with respect to f to be the subset of T ∗X S which | M 1 M | coincide on T ∗U S with φf− charU Y ( 0) for any open subset U and any coherent U Y - | | M D | submodule 0 of U which generates U as a U S-module. M M| M| D | It is clear that charf S( ) is a closed conic analytic subvariety of T ∗X S and that | M |

charf S( )=charf S( )+ψf(X Y T∗Y S). | M | M × |

The functor charf S is additive: | Proposition 2.9 If f : X Y is a relative analytic submersion over S and if −→ 0 0 −→L−→M−→N−→ is an exact sequence of coherent X S-modules then D |

charf S( )=charf S( ) charf S( ). | M | L ∪ | N In the sequel we will need the following lemma essentially due to [8].

13 Proof: In this proof, we always work in some neighborhood of K.

Since admits a good filtration, we can find a coherent X-submodule 0 of M O M M which generates it as a X S-module. Set 0 = X Y 0. By construction, 0 is a D | N D | M N X Y -submodule of which generates it as a X S-module. Obviously, 0 admits a D | M D | N good filtration. Moreover, by definition,

1 φf− charX Y ( 0)=charf S( ). | N | M

The kernel of the canonical X S-linear epimorphism K D |

0 0 X S X Y D | ⊗D | N −→ M −→ is a X S-module which admits a good ﬁltration and we have D |

char ( ) char ( 0)=char ( ). f S f S X S X Y f S | K ⊂ | D | ⊗D | N | M We may thus start over the same construction with replaced by and build the M K requested resolution by induction. 2

Now, by using the graph factorization, we will deﬁne the notion of relative characteristic variety for a map which is not necessarily a relative submersion. Let f : X S Y S be any morphism of relative analytic manifolds. | −→ | Denote by

X X S Y Y −→i × −→q the relative graph factorization of f. First, we notice:

Lemma 2.11 Assume f is a relative submersion and is a coherent X S -module. M D | Then t 1 charf S( )= i0iπ− charq S(i S!( )). | M | | M Hence, for a general f, the following deﬁnition is a natural extension of our previous one.

We also introduce similar deﬁnitions for right X S -modules. D |

14 Note that charf S( ) is a closed conic analytic subset of T ∗X S and that | M | charf S( )=charf S( )+ψf(X Y T∗Y S). | M | M × | A link between the relative characteristic variety and the micro-local theory of sheaves is given in the following theorem:

Theorem 2.13 Let f : X S Y S be a morphism of relative analytic manifolds | −→ | b op over S and assume is an object of Dcoh( X S).Then M D | L 1 SS( X S Y S) p− charf S( ). X S X S M⊗D | D | → | ⊂ | | M Proof: Consider the graph factorization of f:

X X S Y Y. −→i × −→q By Proposition 5.4.4 of [12], if F is a sheaf on X,thenSS(i F) is the natural image of ∗ SS(F). Hence, in view of the deﬁnition of charf S, it is enough to prove the inclusion: | L 1 SS(i!( X S Y S)) p− charq S(i S!( )) X S M⊗D | D | → | ⊂ | | M wherewewritepinstead of pX Y S.Since ×S | L L i!( X S Y S) iS!( ) X Y S Y S, X S X Y S S M⊗D | D | → | ' | M ⊗D ×S | D × | → | we have reduced the proof to the case where f is a relative submersion, what we shall assume now. Since the problem is local on X and

j charf S( )= charf S( ( )), | | M j ZZ H M [∈ we may assume is a coherent right X S-module. Using Lemma 2.10, we are then M D | reduced to consider the case where = 0 , for a coherent right - X Y X S X Y M M ⊗D | D | D | module 0.Now, M L L X S Y S 0 XS YS X S XY M⊗D | D | → | 'M⊗D| D|→| L 1 ( 0 X) 1 f− YS. XY f− Y ' M⊗D| O ⊗ O D| L This last sheaf is locally on X a direct sum of an inﬁnite number of copies of 0 X Y M ⊗D | X . Applying [12] Exercise V.5(i) (which is an easy consequence of Proposition 5.1.1(3) O [loc. cit.]) we get successively

15 2.3 Relative elliptic pairs We shall now define the main object of study of this paper. Let D(X) denote the derived category of the category of sheaves of CI -vector spaces on X and let Db(X) denote the full triangulated subcategory of complexes with bounded amplitude. Recall that a sheaf F of CI-vector spaces is IR-constructible if there is a subanalytic stratification of X along the strata of which Hj (F ) is a locally constant sheaf of finite b rank for any j ZZ. Following [12], we denote by DIR c ( X ) the full triangulated sub- ∈ − category of Db(X) consisting of complexes with IR-constructible cohomology sheaves. b We say for short that an object of DIR c ( X ) is an IR-constructible complex. For such − IR IR an object, SS(F) is a closed subanalytic Lagrangian subset of T ∗X where X denotes X considered with its underlying real analytic manifold structure. We shall identify IR IR T ∗X with (T ∗X) as for example in [12] and simply denote it by T ∗X.Inthispa- per, we will have to consider most of the time the simple dual D0F of F and not its Poincaré-Verdier dual DF . Recall that since X is an oriented topological manifold of dimension 2dX :

D0F = R om(F, CI X)andDF = R om(F, ωX) R om(F, CI X[2dX ]) H H ' H so the two duals coincide up to shift. Since F is constructible, we have the local biduality isomorphism F D 0 D 0 F . −→∼ Deﬁnition 2.14 Let f : X S Y S be a morphism of relative analytic manifolds | −→ | over S.Apair( ,F)isarelative f-elliptic pair if: M b op is an object of Dcoh( X S), •M D | b F is an object of DIR c ( X ), • − 1 p− charf S( ) SS(F) TX∗ X. • | M ∩ ⊂ b op Such a pair is good if moreover is an object of Dgood( X S). Its support is the set M D | supp( ) supp(F ). When f is the canonical map X : X S S S we will say for M ∩ | −→ | shortthat( ,F)isa(good) relative elliptic pair on X S. When S = pt ,wedrop M | { } the word “relative” in the preceding deﬁnitions.

Since charf S( )containscharXS( ), a relative f-elliptic pair is a relative elliptic | M | M pair. Moreover, on a neighborhood of supp , M SS(F) X S T ∗S T ∗ X. ∩ × ⊂ X In particular, an elliptic pair ( ,F )onXis the data of a complex of coherent right M X -modules and an IR-constructible complex F such that D M char( ) SS(F) T ∗ X. M ∩ ⊂ X We shall see in 8 below why this notion is a natural generalization of that of an elliptic § system on a real manifold. There, we will also explain why Theorem 2.15 below may be considered as a generalization of the classical regularity theorem for elliptic systems.

16 Theorem 2.15 Let ( ,F )beanf-elliptic pair. Then the canonical morphism M L L F ( X S Y S) R om(D0F, X S Y S), X S X S ⊗ M⊗D | D | → | −→ H M⊗D | D | → | induced by the morphism F D 0 D 0 F , is an isomorphism. −→ Proof: By [12] Proposition 5.4.14, we know that if G belongs to Db(X)(andFis IR-constructible as above), the natural morphism

F G R om(D0F, G) ⊗ −→ H is an isomorphism as soon as

a SS(F) SS(G) T ∗ X. ∩ ⊂ X Hence, the conclusion follows from Theorem 2.13 by applying the preceding result to

L G = X S Y S. X S M⊗D | D | → | 2

When Y = S,weget:

b op b Corollary 2.16 Let and F be objects of Dcoh( X S) and DIR c ( X ) respectively. M D | − Assume the transversality condition

Deﬁnition 2.17 The dual of a relative pair ( ,F ) is the pair (DX S( ),D0F ). M | M It follows from this deﬁnition that a relative pair is f-elliptic if and only if so is its dual pair.

3Tools

If X is a complex analytic manifold, we have already encountered XIR , the real underlying analytic manifold to X. Here, we shall also make use of X,thecomplex manifold with XIR as underlying space for which the holomorphic functions are the anti-holomorphic functions on X. Recall that X X is a natural complexiﬁcation of × XIR via the diagonal embedding.

17 3.1 Dolbeault complexes with parameters

Let Z and S be complex analytic manifolds and let qZ : Z S Z be the second × −→ projection.

We will denote by Z S S (resp. Z S S , bZ S S ) the sheaf of real analytic functions A × | F × | D × | (resp. inﬁnitely diﬀerentiable functions, distributions) on Z S which are holomorphic × in S. We will also set

ZIR S S =( Z Z SS)ZIR S . D × | D × × | | × IR using the diagonal embedding of Z in Z Z. Locally, operators in ZIR S S are of × D × | the form α β aα,β(z,z,s)Dz Dz Xα,β d Z d S where aα,β(z,z,s) is a section of Z S S ;(z:U CI )and(s:V CI )being A × | −→ −→ holomorphic local coordinate systems on Z and S respectively.

For any Z IR S S -module we will consider the parametric Dolbeault complex D × | M , ·Z· S S( ) A × | M deﬁned by setting p,q 1 p,q ( )=q− 1 Z S S Z Z q− A × | M A ⊗ Z AZ M the formulas for the diﬀerentials being given locally by

p,q p +1,q ∂ : U S S( ) U S S( ) (3.1) A × | M −→ A × | M dZ p,q p,q i p,q a m ∂a m + dz a Dzi m ⊗ 7→ ⊗ i=1 ∧ ⊗ X and

p,q p,q+1 ∂ : U S S( ) U S S( ) (3.2) A × | M −→ A × | M dZ p,q p,q i p,q a m ∂a m + dz a Dzi m ⊗ 7→ ⊗ i=1 ∧ ⊗ X where (z : U CI d Z ) is a holomorphic local coordinate system on Z. Obviously, this −→ deﬁnition is independent on the chosen local coordinate system.

When is equal to Z S S (resp. Z S S , bZ S S ) we will denote the corre- M A × | F × | D , × | , , sponding parametric Dolbeault complex simply by ·Z· S S (resp. Z· · S S , b·Z· S S ). Of A × | F × | D × | course, the natural maps

p p, p, p, ΩZ S S Z· S S Z · S S b Z· S S × | −→ A × | −→ F × | −→ D × | are quasi-isomorphisms.

18 Let be another ZIR S S -module. Using the natural structure of ZIR S S -module N D × | D × | on the sheaf we define the parametric Dolbeault complex of with Z SS N⊗A × | M M coefficients in by the formula N , , · ·( )= ·· ( ). Z SS Z SS N M A × | N⊗A × | M The associated simple complex is the parametric de Rham complex of with coeffi- M cients in .Wedenoteitby ·( ). N N M In this paper, we will only use the preceding notions when is Z S S or bZ S S . N F × | D × | In this case, we have of course

p,q ( )= p,q Z S S Z S S Z S S F × | M F× | ⊗A × | M bp,q ( )= bp,q Z S S Z S S Z S S D × | M D × | ⊗A × | M and the diﬀerentials ∂ and ∂ are given locally by formulas similar to (3.1) and (3.2).

3.2 Realiﬁcation with parameters

Let Z and S be complex analytic manifolds and set n = dZ .ConsiderZ Sas a × relative manifold over S through the second projection .

The parametric realiﬁcation of a left Z S S -module is the sheaf D × | M = . IR S Z S S Z S M | A × | ⊗O × M

In this formula, the ZIR S S -module structure is described locally by the formulas D × |

Dz (a m)=Dza m+a Dzm j ⊗ j ⊗ ⊗ j D(a m)=Da m zj ⊗ zj ⊗ f(a m)=fa m ⊗ ⊗ n where a,f and m are sections of Z S S and respectively; (z : U CI )beinga A × | M −→ local holomorphic coordinate system on Z.

Since Z S S is ﬂat over Z S, parametric realiﬁcation is an exact functor. A × | O × Let us consider the map

δ : Z S Z Z S × −→ × × ( z,s) (z,z,s). 7→ It is clear that 1 δ− ( Z Z S S)= ZIR S S . D × × | D × | Hence the sheaf inverse image by δ of a Z Z S S -module is naturally a ZIR S S - D × × | D × | module. Moreover, one checks easily that

1 1 1 IR S = δ − ( 1 q − )=δ− ( ). Z Z S q − Z S Z M | O × × ⊗ O × M M×O

19 where q : Z Z S Z S is the natural projection and denotes the external × × −→ × × product of -modules. D As usual, using “side changing” functors, we may also define the parametric realification of a right Z S S -module . We still denote it by IR S and check easily D × | M M | that 0 ,n 1 = = δ− ( Ω ). IR S Z S S Z S Z M | A × | ⊗O × M M × Parametric realification is a powerful tool to simplify problems dealing with Z S S - D × | modules thanks to the following result.

Proposition 3.1 Let K be a compact subset of Z and let ∆ be a closed polydisc of S.

Assume is a good Z S S -module. Then, in a neighborhood of K ∆, IR S has a M D × | × M | left resolution by ﬁnite free ZIR S S -modules . D × |

Proof: The assumption insures that Z is a good Z Z S S-module. Hence it is M × O D × × | generated by a coherent Z Z S -module in a neighborhood of the Stein compact subset O × × δ(K ∆) of the complex analytic manifold Z Z S. By Cartan’s Theorem A, it is × × × thus ﬁnitely generated in a neighborhood of δ(K ∆). The conclusion follows easily. × 2

In order to be able to use effectively the preceding proposition in the sequel, we need to understand the links between parametric realification and the finiteness and duality results. These links are made explicit in the following five lemmas. Since the proofs are just easy computational verifications we leave them to the reader. Recall that om· H denotes as usual the internal Hom functor of the category of complexes of sheaves.

0,p Lemma 3.2 a) The sheaf Z S S ( ZIR S S ) is naturally endowed with a structure of A × | D × | left Z S S-module and a structure of right ZIR S S -module and the diﬀerential D × | D × | ∂ is compatible with these two structures.

op 0, b) As a complex of ( Z S S , ZIR S S )-bimodules Z· S S ( ZIR S S ) is quasi-isomor- D × | D × | A × | D × | phic to ( Z S S )IR S [ n ] (where the realiﬁcation uses the right module structure D × | | − of Z S S ) D × | Lemma 3.3 The map

0, 0 , · ( ZIR S S ) Z S S · Z S S Z IR S S Z S S A × | D × | ⊗ D × | F × | −→ F × | ( a 0 ,p Q) u a0,p Qu ⊗ ⊗ 7→ ∧ is an isomorphism of complexes of left Z S S -modules. Combined with the Dolbeault D × | quasi-isomorphism 0 , Z S Z · S S , O × −→ F × | it induces in the derived category the isomorphism

L L Z S ∼ IR S [ n ] Z S S Z S S × | Z IR S S × | M⊗D × | O −→ M − ⊗D × | F

20 for any complex of right Z S S-modules . We have also similar results with D × | M F replaced by b or . D A Lemma 3.4 The map

n, 0 , n,n b · [ n] om· ( · ( Z IR S S ) , b ) Z S S ZIR S S Z S S Z S S D × | − −→ H D × | A × | D × | D × | n,r+n 0, r n,r+n 0, r ϕ [ω − Q (ϕ ω − ) Q] 7→ ⊗ 7→ ∧ · is an isomorphism of complexes of right Z S S -modules. Combined with the Dolbeault D × | quasi-isomorphism n n, ΩZ S S b Z · S S , × | −→ D × | it induces in the derived category the isomorphism

n n,n R om ( , ΩZ S S [n]) ∼ R om ( IR S [ n ] , b ) Z S S ZIR S S Z S S H D × | M × | −→ H D × | M | − D × | for any complex of right Z S S -modules . We have also similar results with b D × | M D replaced by or . A F Lemma 3.5 The natural arrow

n ΩZ· S S ( Z S S ) Ω Z S S [ n ] × | × | × | D −→ 2 n − (resp. bZ· S S ( ZIR S S ) b Z S S [ 2 n ] ) D × | D × | −→ D × | − of complexes of right Y S S (resp. Y IR S S ) modules is a quasi-isomorphism. Together D × | D × | with the relative de Rham quasi-isomorphism

L L n 1 ( Z S) 1 R om ( , ΩZ S [n]) − S [2n] Z S S × − S Z S S M⊗D × | O ⊗ O H D × | M × −→ O

L L  2 n 1  ( IR S Z S S ) 1 R om ( IR S , b Z S S ) − S [2n] | Z IR S S × | − S ZIR S S |  M ⊗D × | F ⊗ O y H D × | M D × | −→ Oy where the horizontal arrows are constructed by contraction followed by the pairings of the preceding lemma, the ﬁrst vertical arrow being the tensor product of the isomorphisms of Lemma 3.3 and 3.4 while the second vertical arrow is the identity.

21 3.3 Trivialization of IR -constructible sheaves In this section, we follow the notations of [12, Ch. VIII]. As in the classical theory of simplicial complexes, the sets U(x) of [loc. cit.] are called open stars. Let us ﬁrst point out some basic facts about the topology of polyhedra.

Lemma 3.7 Let (S, Σ) be a simplicial set and let x S.Then ∈| | (a) y U(x) [x, y] U(x) ∈ ⇒ ⊂ (b) y ∂U(x) [x, y[ U(x) ∈ ⇒ ⊂ where U(x) denotes the open star of x in S . | | Proof: (a) One knows that U(x)= σ. σ σ(x)| | ⊃[ Thus, if y U(x), there is a σ σ(x) such that y σ. Since it is clear that ]x, y] σ ∈ ⊃ ∈| | ⊂| | and that x U(x), one gets that [x, y] U(x). ∈ ⊂ (b) The set σ Σ:σ σ(x) being ﬁnite, one has the equality { ∈ ⊃ } U(x)= σ. σ σ(x)| | ⊃[

Hence, since y ∂U(x), there is a simplex σ σ(x) such that y σ σ.Letσ0 ∈ ⊃ ∈ | |\| | be a simplex included in σ such that y σ0.Ifσ0 σ(x)then]x, y[ σ0 U(x) ∈| | ⊃ ⊂| |⊂ as requested. If σ0 σ(x)thenσ00 = σ0 σ(x) is a simplex of Σ included in σ and 6⊃ ∪ ]x, y[ σ00 U(x) and the conclusion follows. 2 ⊂| |⊂ Lemma 3.8 If (S, Σ) is a simplicial set and if x S then one has the following ∈| | commutative diagram ∂U(x) ]0, 1] /∂U(x) 1 U ( x ) × ×{ } −→∼ ↓↓ ∂U(x) [0, 1] /∂U(x) 1 U ( x ) × ×{ } −→∼ where the horizontal arrows are homeomorphisms, the vertical arrows being the natural inclusions. Proof: Let us deﬁne the continuous application f : ∂U(x) [0, 1] U ( x ) × −→ by setting f(u, t)=(1 t)u+tx. The preceding lemma shows that f(u, t)=f(u0,t0)if − either t = t0 =1or(u, t)=(u0,t0). Moreover, it is clear that for every u U(x)there ∈ is v ∂U(x) such that u [x, v]. From these facts, one deduces that the continuous ∈ ∈ map g : ∂U(x) [0, 1] /∂U(x) 1 U(x) × ×{ }−→ associated to f is bijective. Since ∂U(x) [0, 1] is a compact space, g is an homeomor- × 1 phism. To conclude, it remains to note that f − (U(x)) = ∂U(x) ]0, 1]. 2 ×

22 Proposition 3.9 If (S, Σ) is a simplicial set, then for every open star U(x) of x S ∈| | one has

D0S (CI U(x))=CIU(x) | | Proof: It is clear that

D0S (CI U(x))=Rom(CI U(x), CI S ) | | H | | = RjU(x) (CI U(x)). ∗ It remains to prove that the canonical arrow

CI U(x) RjU(x) (CI U(x)) −→ ∗ is a quasi-isomorphism on U(x). Thanks to the preceding lemma, there is a neighborhood ω of ∂U(x)inU(x) and an homeomorphism

φ : ω ∂U(x) [0,[ −→ × such that φ(ω U(x)) = ∂U(x) ]0,[. We are thus reduced to show that the canonical ∩ × arrows

CI lim H0(V ]0,η[;CI) −→ × V −→,η>0 ∈V 0 lim Hk(V ]0,η[;CI) (k 1) −→ × ≥ V −→,η>0 ∈V are isomorphisms when is a fundamental system of neighborhoods of y ∂U(x). But, V ∈ using homotopy, it is clear that

Hk(V ]0,η[;CI) H k ( V ;CI) × −→∼ and the proof is complete. 2

The following proposition is the main result of this section and will be used as a basic tool in the sequel.

Proposition 3.10 An IR -constructible sheaf F on a real analytic manifold M is quasi- isomorphic to a bounded complex T · of the form

0 CIW CIW CIW 0 a,ia k,ik b,ib ··· −→···ia ⊕Ia −→···ik ⊕Ik −→···ib ⊕Ib −→ ··· ∈ ∈ ∈ where each family (Wk,i )i I is locally ﬁnite, the open subsets Wk,i being subanalytic, k k∈ k k relatively compact, connected and such that

DM0 (CI Wk,i ) CI W , k ' k,ik

23 the diﬀerential dk being such that the induced map T ·

k (d )ji :CIW CI W T · k,i −→ k +1,j k is either 0 if Wk,i Wk+1,j or a complex multiple c IW W of the canonical inclusion 6⊂ ji k+1,j k,i map if Wk,i Wk+1,j. ⊂ Moreover, if F has compact support, we may assume that the set Ik is ﬁnite for every k ZZ. ∈ Proof: From the theory of IR-constructible sheaves, one knows that there is a simpli- 1 cial set (S, Σ)andanhomeomorphismi: S Msuch that i− F is a simplicialy | |−→ constructible sheaf. From a construction due to M. Kashiwara [11] one knows that such k a sheaf is quasi-isomorphic to a bounded complex T · such that each T is a locally ﬁnite direct sum of the sheaves CI U(σ) associated to the open stars of the simplexes of Σ where F is non zero. Since we have just proven in the preceding lemma that for such a sheaf one has

D0(CI U(σ)) CI ' U(σ) the ﬁrst part of the proposition is clear.

Concerning the diﬀerential of the complex, we note that if σ, σ0 are two simplexes of Σ then

Hom(CI U(σ), CI U(σ )) Γ(U(σ); CI U(σ) U(σ )) 0 ' ∩ 0 hence the conclusion since U(σ) is a connected open set. In case F has compact support K, the open stars U(σ) meeting K are in ﬁnite number and since only these stars appear in the components of T ·,thesetsIk are ﬁnite. 2

3.4 Topological S-modules O Let S be a complex analytic manifold. Recall that the sheaf S of holomorphic functions O on S is a multiplicatively convex sheaf of Fr´echet algebras over S (see [7, 25]). Also recall that if V is a relatively compact open subset of a Stein open subset U of X then the restriction map

Γ(U; S) Γ(V ; S) O −→ O is CI -nuclear. From this it follows easily that Γ(U; U )isaFr´echet nuclear (FN) space O and that Γ(V, S) is a dual Fr´echet nuclear (DFN) space. O As in [7], we will consider S as a sheaf of complete bornological algebras and O deal with the category Born( S) of complete bornological modules over S. Recall O O that Houzel has shown that Born( S) has a natural internal hom functor denoted by O ( , ) and an associated tensor product functor denoted by ˆ . They are linked L OS · · · ⊗OS · by the adjunction formula

ˆ Hom Born( )( , )=HomBorn( )( , ( , )). OS M ⊗OS N P OS M L OS N P

24 We denote by L ( , ) the global sections of ( , ) considered as a bornological OS · · L OS · · vector space. For any in Born( S), the functor L ( , ) has a left adjoint. We M O OS M · denote it by ˆ . · ⊗M Following [15], an FN-free (resp. a DFN-free) S-module is a module isomorphic to O E ˆ S for some Fréchet nuclear (resp. dual Fréchet nuclear) space E. It is easily shown ⊗O that the S topological dual ( , S) of an FN-free (resp. a DFN-free) S-module O L OS M O O is DFN-free (resp. FN-free). Moreover both FN-free and DFN-free S-modules are M O S reflexive. O The results needed for the proof of the finiteness, duality and base change theorems for relative elliptic pairs are summarized in the three following propositions. The first one is Corollary 5.1 of [25] and the next two ones are easily deduced from the results in 1–2 of [15] (see also [16]). §

Proposition 3.11 Let · (resp. ·) be a complex of DFN-free (resp. FN-free) S- M N O modules. Assume · and · are bounded from above and M N

u· : · · M −→ N is a continuous S-linear morphism. Assume moreover that u· is a quasi-isomorphism O forgetting the topology. Then · and · have S-coherent cohomology. M N O

Proposition 3.12 Let · be a complex of FN-free S-modules and let be a DFN M O N S-module. Assume · is bounded from above and has S-coherent cohomology. O M + O Then the natural morphism of D ( S) O

( ·, ) R om ( ·, ) L OS M N −→ H OS M N is an isomorphism.

Proposition 3.13 Let · be a complex of FN-free (resp. DFN-free) S-modules and M O let be an FN (resp. DFN) S-module. Assume · is bounded from above and has N O M S-coherent cohomology. Then the natural morphism of D−( S) O O L · · ˆ S M ⊗OS N−→M⊗O N is an isomorphism. In the sequel, when applying the preceding propositions, we will use the following well-known result.

Proposition 3.14 Assume Z, S are complex manifolds. Denote by : Z S S × −→ the second projection. Then, we have the following isomorphisms:

Z S S Γ(Z; Z ) ˆ S ∗F × | ' F ⊗O dZ,dZ dZ ,dZ ˆ ! bZ S S Γc(Z; b ) S = ( Z SS, S). D × | ' D ⊗O LOS ∗F × | O

25 dZ ,dZ Hence, ! bZ S S is a DFN-free S-module which is the topological dual over S of the D × | O O FN-free S-module Z S S .Moreover, O ∗F × |

Z S=Γ(Z; Z)ˆ S. ∗O × O ⊗O Hence, if K is a compact subset of Z,wehave

[( Z S S)K S] Γ(K; Z ) ˆ S ∗ A × | × ' A ⊗O and [( Z S S)K S] is a DFN-free topological S-module. Finally, if T is another ∗ A × | × O complex manifold and p : Z T S S and q : T S S denotes the canonical × × −→ × −→ projections, we have

p Z T S T S Z S S ˆ q T S. ∗F × × | × ' ∗F × | ⊗OS ∗O × 4 Finiteness

4.1 The case of a projection Proposition 4.1 Let Z, S be complex analytic manifolds. Consider Z S as a relative × analytic manifold over S through the second projection .LetGbe an object of b DIR c ( Z ) and set F = G CI S. Assume that ( ,F ) is a good relative elliptic pair with − × M -propersupportonZ SS.Then × | L R!(F Z S) Z S S ⊗M⊗D × | O × b is an object of D ( S). coh O Proof: By “dévissage”, it is obviously sufficient to prove the result when is a M Z S S -module which admits a good filtration on a neighborhood of any compact subset D × | of Z S. × It follows from the relative regularity theorem (Theorem 2.15) that the canonical map L L F Z S R om(D0F, Z S) Z S S Z S S ⊗M⊗D × | O × −→ H M⊗D × | O × is an isomorphism. Using Lemma 3.3, we get the isomorphism

L R (F IR S Z S S ) (4.1) ∗ | Z IR S S × | ⊗M ⊗D × | A L R R om(D0F, IR S Z S S ) . ∗ | Z IR S S × | −→ H M ⊗D × | F 1 Let V be the interior of a closed polydisc ∆ of S. Since supp( ) supp(F ) − (∆) M ∩ ∩ is compact, we can ﬁnd a compact subset K in Z such that K V is a neighborhood 1 × of supp( ) supp(F ) − (V ). Replacing S by V and G by GK shows that we may M ∩ ∩ assume from the beginning that G has compact support. Moreover, by Proposition 3.1 we may also assume that IR S is quasi-isomorphic to complex · whose components M | L

26 are free ZIR S S -modules of ﬁnite rank. Since IR S has bounded amplitude, we may D × | M | even assume k =0fork 0. L We know by Proposition 3.10 that D0G is isomorphic to a bounded complex T · of the form

CIWk,i ···−→i ⊕Ik −→··· ∈ where W is relatively compact subanalytic subset of Z such that D (CI )=CI . k,i 0 Wk,i Wk,i Thus G D0D0G is quasi-isomorphic to a complex C· of the form ' CI Wk,i ···−→i ⊕Ik −→··· ∈

It is clear that the sheaf ( Z S) K S (resp. ( Z S ) U S) is acyclic for the the functor A × | × F × | × K S (resp. U S ) for any compact subset K (resp. any open subset U)ofZ. Hence | × | × we may∗ view isomorphism∗ (4.1) more explicitly as the morphism

((C· CI S) · Z S S ) (4.2) ∗ ZIR S S × ⊗L ⊗D × | A × | om(T · CI S, · Z S S ) ∗ ZIR S S −→ H × L ⊗D × | F × | in the category of complexes of S-modules (not the derived category). Let us denote O by · (resp. · ) the source (resp. target) of the preceding arrow. R1 R2 The components of · (resp. · ) are easily seen to be ﬁnite sums of the sheaves R1 R2

W S ( Z S S ) (resp. Wk,i S ( Z S S )). k,i × | W k,i S | × × | Wk,i S | × ∗ A | × ∗ F | ×

Hence, · (resp. · ) is naturally a complex of DFN-free (resp. FN-free) topological R1 R2 S-modules. For these natural topologies, the regularity quasi-isomorphism is clearly O continuous. Applying Proposition 3.11 we conclude that · has S-coherent cohomol- R2 O ogy and the proof is complete. 2

4.2 The general case Theorem 4.2 Let f : X S Y S be a morphism of relative analytic manifolds over | −→ | S. Assume ( , F ) is a good relative f-elliptic pair with f-proper support; i.e. M b op is an object of Dgood( X S), •M D | b F is an object of DIR c ( X ) , • − 1 φ − charf S( ) SS(F) TX∗ X, • | M ∩ ⊂ supp( ) supp(F ) is f-proper. • M ∩ Then f ( F)is an object of Db ( op ). S! good Y S | M⊗ D |

27 Proof: By “dévissage”, it is obviously sufficient to prove the result when is a X S- M D | module which admits a good filtration on a neighborhood of any compact subset of X. Decomposing f through its graph embedding i : X X Y −→ × shows that it is sufficient to prove the finiteness theorem for the second projection

p2 : X Y S Y S × | −→ | b op b and the pair i S!( ) Ob(Dgood( X Y S)), F CI Y Ob(DIR c ( X Y )) since by the | M ∈ D × | × ∈ − × projection formula we have

i S!( ) (F CI Y ) i S!( F). | M ⊗ × ' | M⊗ From the deﬁnition of charf S( ), it is clear that | M

charp2 S(i S!( )) SS(F CI Y ) | | M ∩ × is contained in the zero section of T ∗(X Y S). So if Y = S, the theorem is a × | consequence of the results obtained in the case of a projection. To conclude, we will show that if f is a relative submersion and the theorem is true for f : X Y Y Y then it is also true for f : X S Y S . | −→ | | −→ | We will use a device introduced in [8] and extended in Lemma 2.10. Let ∆ be a polydisc in Y and denote by K the compact subset of X deﬁned by

1 K = supp( ) SS(F) f− (∆). M ∩ ∩ Using Lemma 2.10, it is easy to see that, in a neighborhood of K, is isomorphic M to a complex of right -modules of the form where is a coherent X S X Y X S D | R⊗D | D | R right X Y -submodule which admits a good filtration and is such that D | 1 φ− charX Y ( ) charf S( ). | R ⊂ | M Moreover, this complex may be assumed to be bounded from above. f has finite cohomological dimension, it is thus sufficient to Since the functor S! prove the coherence on| ∆ of the cohomology of f (F )when has the special S! | ⊗M M form = 0 where 0 is a coherent -module which admits a good X Y X S X Y M M ⊗D | D | M D | filtration.

In this case, one knows that the complex f (F 0)has Y-coherent cohomology, Y ! ⊗M O and the chain of isomorphisms |

L Rf!(F X S Y S) X S ⊗M⊗D | D | → | L L = Rf!(F ( 0 X S) X S Y S) X Y X S ⊗ M ⊗D | D | ⊗D | D | → | L Rf!(F 0 X S Y S) ∼ X Y −→ ⊗M ⊗D | D | → | L L Rf!(F 0 X) Y S ∼ X Y Y −→ ⊗M ⊗D | O ⊗O D | shows that f (F ) belongs to Db ( ). 2 S! good Y S | ⊗M D |

28 Corollary 4.3 In the situation of the preceding theorem, the well known formula:

L L f (F ) Y Rf!(F X) S! Y S ∼ X S | ⊗M ⊗D | O −→ ⊗M⊗D | O gives the inclusion:

t 1 (f (F )) f f − char ( ) charY S S! π 0 X S | | ⊗M ⊂ | M Proof: By Theorem 2.13 and Proposition 5.4.4 and 5.4.14 of [12], we know that

1 pX− S charf S( ) SS(F) TX∗ X. | | M ∩ ⊂ Moreover, one has:

1 t 1 pX− ScharX S( )+ f0(X Y T∗Y) pX− Scharf S( ). | | M × ⊂ | | M Hence,

5 Duality

Let f : X S Y S be a morphism of relative complex manifolds. Our aim in this | −→ | section is to prove that, under suitable hypotheses, duality commutes with direct images (see Theorem 5.15 for a precise statement). The proof will use the graph decomposition of f and various “d´evissages”. Hence, it is necessary to construct ﬁrst the natural transformation: f D D f S! X S Y S S ! | ◦ | −→ | ◦ | and to check its compatibility with respect to composition in f. This will be a consequence of the explicit construction of the trace morphism for -modules given in the D next section. We follow the lines of [24] (see also [11, 17]).

29 5.1 The relative duality morphism Recall that if f : X Y is a holomorphic map then it induces integration maps −→

p,q p+dX ,q+dX p + d Y ,q+dY f : f! bX b Y ∗ D −→ D commuting with the Dolbeault operators. We will use this fact and the machinery of distributional Dolbeault complexes of 3.1 to construct canonically the duality map for § right -modules. D Let be a left X -module. To simplify notations, we will set M D , , bX· · ( )= b·X· pt pt ( IR pt ). D M D ×{ }|{ } M |{ } Hence, the components are

p,q p,q bX ( )= bX D M D ⊗OX M and the diﬀerentials are given in a local coordinate system z : U CI d Z by −→ p,q p,q p+1,q ∂ : bX bX D ⊗OX M−→D ⊗OX M dX p,q u P ∂ u P + dzi u Dzi P ⊗ 7→ ⊗ i=1 ∧ ⊗ X and

∂p,q : bp,q bp,q+1 X X X X D ⊗O M−→Dp,q ⊗O M u P ∂ u P ⊗ −→ ⊗ respectively. Also recall that we denote by bX· ( ) the simple complex associated , D M with b· · ( ). D X M

Lemma 5.1 The diﬀerential of bX· ( X ) is compatible with the right X -module D D op D structure of its components and, in Db( ), one has a canonical isomorphism: DX

b· ( X )[dX] Ω X . D X D −→∼

Proof: The compatibility of the diﬀerential of b· ( X )withtheright X-module D X D D structure of its components is a direct consequence of the local forms of ∂ and ∂ recalled above. p Using the fact that X is ﬂat over X and the Dolbeault resolution of Ω ,weget D O X the quasi-isomorphisms

p p, p, ΩX X ∼ b X· X ∼ b X·( X ). ⊗OX D −→ D ⊗OX D −→ D D Hence, Weil’s lemma shows that the natural morphism

DR· ( X ) b · ( X ) X D −→ D X D

30 from the holomorphic to the distributional de Rham complex of X is a quasi-isomorph- D ism of complexes of right X -modules and the conclusion follows from the Spencer D quasi-isomorphism

DR· ( X ) ΩX [ dX]. X D ' − 2

Lemma 5.2 To any morphism f : X Y of analytic manifolds one can associate a −→ canonical integration morphism

f : f! bX· ( X Y )[2dX ] b Y· ( Y )[2dY ]. ∗ D D → −→ D D in the category of bounded complexes of right Y -modules. D Proof: At the level of components the integration morphism is obtained as the following chain of morphisms:

p+dX ,q+dX p + d X ,q+dX 1 f! b ( X Y ) ∼ f ! ( b X f 1 f − Y ) D D → −→ D ⊗ − OY D p + d X ,q+dX ∼ f ! b X Y −→ D ⊗OY D p + d Y ,q+dY b Y Y −→ D ⊗OY D p + d Y ,q+dY b ( Y ). −→∼ D Y D To get the second morphism one has used the projection formula, the fact that bX is D a soft sheaf and the fact that Y is locally free over Y . The third arrow is deduced D O from the integration of distributions along the fibers of f. To conclude, we need to show that the integration morphism is compatible with the differentials of the complexes involved. Thanks to the local forms of the differentials, this is an easy computational verification and we leave it to the reader. 2

Lemma 5.3 If f : X Y and g : Y Z are morphisms of complex analytic −→ −→ manifolds, one has the following commutative diagram: 1 g!(f! bX· ( X Y )[2dX ] Y Z ) g ! ( b Y· ( Y )[2dY ] Y Z ) D D → ⊗DY D → −→ D D ⊗DY D → 2 3  4  g!f! bX· ( X Z)[2dX ] b Z· ( Z )[2dZ]. D Dy → −→ D Dy In this diagram, arrow (1) is deduced by tensor product and proper direct image from f , arrow (2) is an isomorphism deduced from the projection formula, arrow (3) is g ∗ ∗ and arrow (4) is equal to (g f) . ◦ ∗ Proof: Going back to the deﬁnition of the various morphisms, one sees easily that the commutativity of the preceding diagram is a consequence of the Fubini theorem for distributions, that is, the formula (g f) (u)=g(f(u)) ◦ ∗ ∗ ∗ where g and f denotes the push-forward of distributions along g and f respectively, ∗ ∗ u being a distribution with g f proper support. 2 ◦

31 Proposition 5.4 If f : X Y is a morphism of complex analytic manifolds then −→ there is a canonical integration arrow

: f (ΩX [dX ]) Ω Y [ d Y ] f ! −→ R in Db( op).Moreover,ifg:Y Z is a second morphism of complex analytic DY −→ manifolds then = g ( ) g f g ! f ◦ ◦ R R R Proof: One gets the arrow f by composing the morphisms: R f (Ω [d ]) Rf (Ω [d ] L ) ! X X ∼ ! X X X Y −→ ⊗DX D → ∼ Rf!( bX· ( X Y )[2dX ]) −→ D D → ∼ f ! ( b X· ( X Y )[2dX ]) −→ D D → b · ( Y )[2dY ] −→ D Y D Ω Y [ d Y ] −→∼ Let us point out that the second and last isomorphisms come from Lemma 5.1, that p,q the third one is deduced from the fact that bX ( X Y ) is c-soft and that the fourth D D → arrow is given by Lemma 5.2. The compatibility of integration with composition is then a direct consequence of Lemma 5.3. 2

Corollary 5.5 If f : X S Y S is a morphism of relative analytic manifolds over S | −→ | then there is a canonical arrow

: f (Ω [d ]) Ω [ d ] . f S S! X S X S Y S Y S | | | | −→ | | R Moreover, if g : Y S Z S is another morphism of relative analytic manifolds over | −→ | S then = g ( ) g f S g S S! f S ◦ | | ◦ | | Proof: Using the canonical morphismR R R

L L ΩX [dX] X Y Ω X [ d X ] X Y X S X ⊗D | D → −→ ⊗D D → and the integration morphism

L f : Rf!(ΩX[dX ] X Y ) Ω Y [ d Y ] ⊗DX D → −→ R we get the morphism

L Rf!(ΩX [dX ] X Y ) Ω Y [ d Y ] . X S ⊗D | D → −→ Since 1 X Y XS Y S 1 f− Y f− Y S D → 'D | → | ⊗ D | D

32 1 op as a ( X S,f − Y )-bimodule and ΩY is a right Y -module, we get a Y S-linear mor- D | D D D | phism: L Rf!(ΩX [dX ] X S Y S) Ω Y [ d Y ] . X S ⊗D | D | → | −→ 1 1 Tensoring on both sides by − Ω⊗− [ dS] and using the projection formula, we get the Y S − requested relative integration map:

L Rf!(ΩX S[dX S] X S Y S) Ω Y S [ d Y S ] . X S | | ⊗D | D | → | −→ | | The last part of the corollary is then an easy consequence of the similar result for S = pt . 2 { }

Deﬁnition 5.6 One deﬁnes the direct image of a right ( X S, X S)-bimodule by D | D | M setting:

L L f ( )=Rf!(( X S Y S) X S Y S) S! X S | → | X S | → | | M M⊗D | D ⊗D | D Lemma 5.7 There is a canonical isomorphism

L L [( X S X S) X S Y S] X S Y S X X S X S D | ⊗O D | ⊗D | D | → | ⊗D | D | → | 1 ∼ X S Y S f 1 f − Y S −→ D | → | ⊗ − O Y D | compatible both with the structure of left X S-module and the structure of right 1 1 D | (f − Y S ,f− Y S)-bimodule. D | D | Proof: One has the chain of isomorphisms

L L [( X S X S) X S Y S] X S Y S X X S X S D | ⊗O D | ⊗D | D | → | ⊗D | D | → | L [ X S X S Y S ] X S Y S ∼ X X S −→ D | ⊗O D | → | ⊗D | D | → | L ∼ X S Y S X S Y S −→ D | → | ⊗O X D | → | 1 ∼ X S Y S f 1 f − Y S | → | − Y | −→ D ⊗ O D1 1 X S Y S 1 ( f − Y S 1 f − Y S ) ∼ f − Y S f − Y −→ D | → | ⊗ D | D | ⊗ O D | 1 1 X S Y S 1 ( f − Y S 1 f − Y S ) ∼ f − Y S f − Y −→ D | → | ⊗ D | D | ⊗ O D | In the second isomorphism we have used the exchange lemma. In the fourth line, the 1 1 last tensor product uses the structure of left f − Y -module of f − Y S. In the ﬁfth O 1D | isomorphism the last tensor product uses the structure of right f − Y -module of the 1 O ﬁrst factor and the structure of left f − Y -module of the second one. Finally, in the O last line, we have used the exchange lemma again. 2

Proposition 5.8 Let be a right X S-module and let X S be the associated M D | M⊗OX D | right X S-bimodule. If f : X Y is a morphism of relative analytic manifolds over D | −→ S then one has the following canonical isomorphism

L L f ( X S) ∼ f ( ) Y S S! X | S ! Y | | M⊗O D −→ | M ⊗O D op op in the derived category D( Y S Y S). D | ⊗D |

33 Proof: This is a direct consequence of the preceding lemma. 2

Deﬁnition 5.9 The diﬀerential trace map associated to a morphism

f : X S Y S | −→ | of relative analytic manifolds over S is deﬁned to be the arrow

trf : f X S Y S S! | | | K −→ K op op in the derived category D( Y S Y S) obtained by composing the following arrows: D | ⊗D | L f X S = f (ΩX S[dX S] X S) S! | S! | | X | | K | ⊗O D L ( f Ω X S [ d X S ]) Y S ∼ S ! Y −→ | | | ⊗O D | L ∼ Ω Y S [ d Y S ] Y S −→ | | ⊗O Y D | where the ﬁrst arrow comes from the deﬁnition of X S (see p. 11), the second one K | being a consequence of the preceding proposition and the third one being constructed by tensor product with the integration arrow of Corollary 5.5. By construction, trf is compatible with the composition of maps.

Proposition 5.10 Assume f : X S Y S is a morphism of relative analytic mani- | −→ | folds over S. Then the diﬀerential trace map

Proof: Since, by deﬁnition,

D ( )=R om ( , X S) X S X S | M H D | M K | we have a canonical morphism

R om (f ,f X S) R om (f , Y S ) Y S S! S! | Y S S! | H D | | M | K −→ H D | | M K associated to trf gives the requested duality morphism. The construction shows that it is natural in . The compatibility with composition in f comes from the corresponding M property of trf . 2

34 To conclude this section, we will show that the diﬀerential duality morphism is compatible with the duality morphism of complex analytic geometry.

Recall that since X S is an X -module, we have a well deﬁned scalar extension D | O functor

op EX S :Mod( X) Mod( X S) | O −→ D | XS. F7→F⊗OXD|

The image by this functor of an X -module is coherent as a right X S-module if O F D | and only if is coherent as an X-module. For a coherent X-module ,onesets F O O F

DXS( )=R om ( , ΩX S[dX S]) | F H OX F | | hence, we have the canonical isomorphism:

(Rf! ) Y S f ( XS). Y ∼ S ! X F ⊗O D | −→ | F⊗O D | With these facts in mind, we can now state:

Proposition 5.11 Let f : X S Y S be a morphism of relative analytic manifolds. | −→ | Assume is a coherent X-module. Then we have the commutative diagram: F O

Rf!DX S ( ) Y S D Y S ( Rf! ) Y S | F ⊗OY D | −→ | F ⊗OY D |

f D (  ) D ( f (  )) S! X S  X XS Y S S !  X XS | | F⊗y O D | −→ | | F⊗y O D | where the ﬁrst and second horizontal arrows come respectively from the geometric and diﬀerential duality morphisms, and the vertical arrows isomorphisms are deduced from the compatibility with direct image and duality of the scalar extensions functors EX S | and EY S . | Proof: Consider the morphism

ΩX S[dX S] X S (5.1) | | −→ K | ω ω 1X S. 7→ ⊗ | It follows easily from the deﬁnition of the diﬀerential integration map that we have the following commutative diagram:

Rf!ΩX S[dX S] Ω Y S [ d Y S ] | | −→ | |

f  X S Y S . S !  | | | Ky −→ K y

35 In the preceding diagram the horizontal arrows are the geometric and diﬀerential trace maps, the ﬁrst vertical arrow is deduced from (5.1) by using the canonical section

1X S Y S of X S Y S and the second vertical arrow is (5.1) with X replaced by Y . | → | D | → | Since the geometric and diﬀerential duality morphisms are directly constructed from the corresponding trace maps the result is easily reduced to the commutativity of the preceding diagram. 2

5.2 The case of a closed embedding Proposition 5.12 Let i : X S Y S be a closed relative embedding. Then, for | −→ | every coherent right X S-module , the canonical morphism D | M

i S!DX S( ) D Y S ( i S ) | | M −→ | | ∗ M is an isomorphism.

Proof: Since the problem is local on X, we may assume has a bounded resolution M by ﬁnite free right X S -modules. Thus it is suﬃcient to prove the result for = X S . D | M D | Since we have

X S = X X S X S D | O ⊗D | D | it follows from Proposition 5.11 that the result is a direct consequence of the corresponding result for -modules. Since we do not have a precise direct reference for this O well known result we recall it in the following lemma. 2

Lemma 5.13 If i : Z X is a closed embedding of analytic manifolds, then for any b −→ b object of D ( Z) the complex Ri! is an object of D ( X) and the geometric F coh O F coh O duality morphism

Ri!R om ( , ΩZ [dZ ]) R om (Ri! , ΩX [dX]) H OZ F −→ H OX F is an isomorphism.

Proof: Since the result is of local nature and the duality morphism is compatible with composition, it suﬃcient to consider the case when = Z and F O

i : U 0 U 0 U 00 −→ × z0 (z0, 0) 7→

dZ where U 0 (resp. U 00)isanopenneighborhoodof0inCI (resp. CI). In this case, the arrow

i!ΩZ [dZ] R om (i! Z , ΩX [dX]) −→ H OX O corresponds up to shift to the arrow

i!ΩZ R om (i! Z , ΩX [1]) −→ H OX O

36 deduced from the arrow

dX , i!ΩZ om (i! Z , bX ·[1]) (5.2) −→ H OX O D i!ω (i!h i (hω)) 7→ 7→ ∗ by using the natural map from om to R om and the Dolbeault resolution X X d X , H O H O ΩX ∼ b X · . Using the negative Koszul complex K (z00, X ) as a free resolution of −→ D · O i! Z, the map (5.2) corresponds to the morphism of complexes O

dX , i!ΩZ om O (K (z00, X), bX ·) (5.3) −→ H X · O D which associates to i!ω the morphism of complexes which sends h to i (h Zω)indegree ∗ | zero and is zero in other degrees. The target of the preceding arrow is the simple complex associated to the double , complex K· · below

bdX ,0 b d X , 1 bdX,dX D X −→ D X ··· −→ D X z z z 00 00 ··· 00 xdX ,0 xd X , 1 dxX,dX bX bX bX D  −→ D  ··· −→ D  where the horizontal maps are the ∂ Dolbeault operators, the vertical ones being multiplication by z00. In the preceding diagram the term of bidegree (0, 0) is in the upper left corner and the image of i!ω by the arrow (5.3) corresponds to the section i ω of ∗ dX ,1 bX in bidegree ( 1, 1). D − , Since the canonical inclusion of K (z00, ΩX ) in the simple complex sK· · is a quasi- · , isomorphism, it follows that the cohomology of sK· · is concentrated in degree zero and 0 , that H (sK· ·) is isomorphic to ΩX /z00ΩX . Nowwehavesuccessively

iω = ω(z0) δ(z00)dx00 dy00 ∗ ∧ ∧ dz00 = ω(z0) ∂ ∧ 2iπz00 !

dz00 = ∂ ω(z0) ∧ 2iπz00 !

0 , and this shows that i ω has the same cohomology class in H (sK· ·) as the section ∗0,0 0 ω(z0) (dz00/2iπ)ofK . Hence the arrow (**) corresponds at the level of H to the ∧ isomorphism

i!ΩZ Ω X /z00ΩX −→ dz00 i!ω ω(z0) 7→ " ∧ 2iπ # z00ΩX and the conclusion follows. 2

37 5.3 The case of a projection As for the ﬁniteness theorem our starting point will be the case of a projection.

Proposition 5.14 Let Z, S be complex analytic manifolds and denote by n the complex dimension of Z.ConsiderZ Sas a relative analytic manifold over S through × b the second projection .LetGbe an object of DIR c ( Z ) and set F = G CI S . Assume − × that ( ,F ) is a good relative elliptic pair with -proper support on Z S S. Then the M × | natural pairing

L L n R ( F Z S) R!(D0F R om ( , Ω [n])) S Z S S S Z S S Z S S ∗ M⊗ ⊗D × | O × ⊗O ⊗ H D × | M × | −→ O identiﬁes each complex with the S dual of the other. O Proof: Since the dual of a relative elliptic pair is a relative elliptic pair, we need only to show that the map

n R!(D0F R om ( , Ω [n]) Z S S Z S S ⊗ H D × | M × | L R om (R (F Z S), S) S Z S S −→ H O ∗ ⊗M⊗D × | O × O deduced from the duality pairing is an isomorphism in the derived category. Using Lemmas 3.3, 3.4 and 3.6 and the regularity quasi-isomorphism, it is equivalent to prove that the canonical map

n,n R!(D0F R om ( IR S , b )) (5.4) ZIR S S Z S S ⊗ H D × | M | D × | L R om (R R om(D0F, IR S Z S ) , S )) S ∗ | Z IR S S × −→ H O H M ⊗D × | F O is an isomorphism in the derived category. We will work as in the proof of Proposition 4.1 and use the notations introduced there. Using the resolution · of IR S and the resolution T · of D0G, we will compute L M | explicitly the preceding morphism. We already know that

L R R om(D0F, IR S Z S ) ∗ | Z IR S S × H M ⊗D × | F ∼ om(T · CI S, · Z S )= 2·. ∗ ZIR S S × −→ H × L ⊗D × | F R

Since bWk,i S is acyclic for the functor W S ,weget D × | k,i× ! n,n R!(D0F R om ( IR S , b Z S S )) ⊗ H DZIR S S M | D × | × | n,n ∼ ! ((T · CI S) om ( · , b )). ZIR S S Z S S −→ × ⊗H D × | L D × |

We denote by · this last complex. R3 The components of · (resp. · ) are ﬁnite sums of the sheaves R2 R3 ( ) (resp. ( bn,n )) Wk,i S Wk,i S Wk,i S! Wk,i S | × ∗ F × | × D ×

38 which are FN-free (resp. DFN-free) S-modules. Hence, · and · are naturally O R2 R3 complexes of topological S-modules. O For any open subset U of Z, we know that

bn,n = ( , ) U S S ! U S S S U S S U S S S × | D × | L O | × | ∗F × | O where the second member of the preceding equality is the sheaf of continuous S-linear O homomorphisms between the FN-free S-module U S S U S S and S. Hence we get O | × | F × | O the canonical isomorphism ∗ k k 3 ∼ ( 2− , S ) R −→ L O S R O for any integer k. One checks easily that these maps deﬁne an isomorphism of complexes

3· ∼ ( 2· , S ) . R −→ L O S R O Moreover, the composition of this morphism with the natural morphism

( 2· , S) R om ( 2· , S) (5.5) L OS R O −→ H OS R O gives the map (5.4).

Since · has S-coherent cohomology, Proposition 3.12 shows that (5.5) is a quasi- R2 O isomorphism and the proof is complete. 2

5.4 The general case Theorem 5.15 Let f : X S Y S be a morphism of relative analytic manifolds | −→ | over S. Assume ( , F ) is a good relative f-elliptic pair with f-proper support; i.e. M b op is an object of Dgood( X S), •M D | b F is an object of DIR c ( X ) , • − 1 φ − charf S( ) SS(F) TX∗ X, • | M ∩ ⊂ supp( ) supp(F ) is f-proper. • M ∩ Then the duality morphism

f (D F D ( )) D ( f ( F )) S! 0 X S Y S S ! | ⊗ | M −→ | | ⊗M is an isomorphism.

Proof: Using the factorization of f through its graph embedding

i : X X Y −→ × we deduce from the results obtained for closed embeddings that the theorem will be true if it is true for the second projection

q : X Y S Y S × | −→ |

39 and the pair

b op b (i S!( ),F CIY) Ob(Dgood( X Y S) DIR c ( X Y )). | M × ∈ D × | × − ×

From the deﬁnition of charf S( ), it is clear that | M

charq S(i S!( )) SS(F CI Y ) | | M ∩ × is in the zero section of T ∗(X Y S). So if Y = S, the theorem is a consequence of the × | results obtained in the product case. To conclude, we will show that if f is a relative submersion and the theorem is true for f : X Y Y Y then it is also true for f : X S Y S . | −→ | | −→ | Let us assume ﬁrst that there is a coherent right X Y -module 0 such that D | M

= 0 . X Y X S M M ⊗D | D | One get successively:

DY S (f (F )) | ! ⊗M L L R om ([Rf!(F 0 X)] Y S , Y S) ∼ Y S X Y Y −→ H D | ⊗M ⊗D | O ⊗O D | K | R om (f (F 0), Y) ∼ Y Y ! Y Y S −→ H O | ⊗M O ⊗O K | f ( D 0 F D ( 0 )) ∼ Y ! X Y Y Y S −→ | ⊗ | M ⊗O K | L 1 Rf!(D0F R om ( 0, X Y ) X 1 f − Y S) ∼ X Y X Y f − Y −→ ⊗ H D | M K | ⊗D | O ⊗ O K | L Rf!(D0F R om ( 0, X S) X S Y S) ∼ X Y X S −→ ⊗ H D | M K | ⊗D | D | → | f ( D 0 F R om ( 0 , )) ∼ S ! X S X Y X S X S −→ | ⊗ H D | M ⊗D | D | K | f ( D F D ( )) ∼ S ! 0 X S −→ | ⊗ | M and the theorem is proved. The general case is reduced to the preceding case by using Lemma 2.10 as in the proof of Theorem 4.2.

6 Base change and K¨unneth formula

6.1 Base change

Recall that to any morphism b : Sb S of complex manifolds is associated a base −→ change functor

( )b :Man(S) Man(Sb) · −→ X S X S S b S b | −→ × |

40 which transforms relative manifolds over S into relative manifolds over Sb.Theaimof this section is to study the behavior of relative elliptic pairs under this functor. The main result is Theorem 6.5.

Let us ﬁx a base change map b : Sb S . For any relative manifold X S,wedenote −→ | by Xb Sb its image by the base change functor ( )b and by bX the projection from Xb to | · X. By construction, we have the cartesian square:

X X S −→ b X 2 b

Xxb Sxb .    −→X b  Hence, there is a canonical ring morphism

1 bX− X S X S D | −→ D b | b and we may introduce the following deﬁnition.

Deﬁnition 6.1 The base change functor for relative right -modules is the functor D op op D( X S) D ( X S ) D | −→ D b| b 1 L bX− 1 Xb Sb. bX− X S M7→ M⊗ D | D | b op b op This functor clearly induces a functor from Dgood( X S)toDgood( X S ). D | D b| b The base change functor for sheaves of CI -vector spaces is the functor

D(X) D ( X b ) −→ 1 F b− F. 7→ X b b This functor clearly induces a functor from DIR c ( X )toDIR c ( X b ). Since the context − − will avoid any possible confusion, we denote all these functors by ( )b. · Let us consider now a morphism f : X S Y S of relative manifolds. We denote | −→ | by fb : Xb Sb Y b S b the image of f by the base change associated with b.Onechecks | −→ | easily that the square: f X Y −→ b X b Y (6.1) x x X b Yb  −→f b  is cartesian.

Proposition 6.2 Using the notations introduced above: a) There is a canonical morphism

1 L 1 bX− X S Y S 1 1 fb− Yb Sb X b S b Y b S b fb− bY− Y S D | → | ⊗ D | D | −→ D | → | b 1 1 op in D (bX− X S fb− Y S ). D | ⊗ D b| b

41 b) The preceding morphism induces a natural morphism (f ( )) f ( ) S! b b S ! b | M −→ | b M op for in D( X S). M D | c) The morphisms in (a) and (b) are isomorphisms if either f or b is a closed embedding. Proof: Since

1 L 1 1 L 1 bX− X S Y S 1 1 fb− Yb Sb ∼ b X− X 1 1 f b− Y b S b fb− bY− Y S b X− f − Y D | → | ⊗ D | D | −→ O ⊗ O D | and 1 1 f − Xb Sb Yb Sb ∼ X b f − b Y b S b D | → | −→ O ⊗ b O Y b D | 1 the canonical morphism bX− X X b induces the morphism in (a). op O −→ O For any in D( X S), we construct the morphism in (b) as the chain of morphisms: M D | 1 L L (f )b = bY− Rf!( X S Y S) 1 Yb Sb S! X S bY− Y S | M M⊗D | D | → | ⊗ D | D | 1 L L ∼ Rfb!bX− ( X S Y S) 1 Yb Sb X S bY− Y S −→ M⊗D | D | → | ⊗ D | D | 1 L 1 L 1 ∼ Rfb!(bX− 1 bX− X S Y S 1 1 fb− Yb Sb) bX− X S fb− bY− Y S −→ M⊗ D | D | → | ⊗ D | D | 1 L Rfb!(bX− 1 Xb Sb Yb Sb) bX− X S −→ M⊗ D | D | → | f ( ) . ∼ b S ! b −→ | b M This chain of morphisms is obtained using the definition of the base change functor, the fact that the square (6.1) is cartesian, the projection formula, the morphism constructed in (a) and again the definition of the base change functor. To conclude the proof, it is sufficient to show that if either f or b is a closed embedding then the morphism constructed in (a) is an isomorphism. Assume f is a closed embedding. The problem being local, we may assume there are open neighborhoods U and V of zero in CI n and CI m respectively with X = U S, × Y = U V S and × × f : U S U V S × −→ × × ( u, s) ( u, 0,s). −→ Then, we get Xb = U Sb, Yb = U V Sb and × × × fb : U Sb U V S b × −→ × × ( u, sb) ( u, 0,sb). −→ Moreover, bX (u, sb)=(u, b(sb)) and bY (u, v, sb)=(u, v, b(sb)). In this simple geometric situation, we have the Koszul quasi-isomorphisms:

K ( Y ; v1,...,vn) ∼ f X · O −→ ∗ O K ( Y b ; v 1 b Y ,...,vn bY) ∼ f b X b · O ◦ ◦ −→ ∗ O

42 where v1,...,vn denotes the functions on Y induced by the standard coordinates on V . Hence, we get the isomorphisms:

1 L 1 1 b− X 1 1 f − f − K ( ; v1 bY ,...,vn bY) X b− f b Yb Sb b Yb Sb O ⊗ X − OY D | ' · D | ◦ ◦ L 1 1 X 1 f− Y S f− K( Y S ;v1 bY,...,vn bY) b f− b b b b b b O ⊗ b OYb D | ' · D | ◦ ◦ and the conclusion follows. The case where b is a closed embedding is treated in a similar way. 2 The following easy lemma will be useful in the sequel. We leave its proof to the reader.

Lemma 6.3 Let f : X S Y S be a relative submersion and let b : Sb S be a | −→ | −→ base map. Consider X as a relative manifold over Y through the map f and assume op is an object of D( X Y ).Then N D |

fb:XbYb Y b Y b | −→ | is the image of f : X Y Y Y by the base change associated with bY and | −→ |

( )b b . XY XS Y X Y Xb Sb N⊗D | D | 'N ⊗D b| b D | The behavior of the characteristic variety under base change is given by the following result.

Proposition 6.4 Let f : X S Y S be a morphism of relative manifolds. In the | −→ | diagram

T ∗Xb Sb X b X T ∗ X S T ∗ X S t | ←−( b X ) 0 × | (−→b X ) π | b op the ﬁrst arrow is an isomorphism and for any object of Dcoh( X S) we have M D | t 1 charf S ( b) (bX )0(bX )π− charf S( ). b| b M ⊂ | M Proof: Using the graph factorization of f and part (c) of Proposition 6.2 we are reduced to the case where f is a relative submersion. In this case, assume is generated as M aright XS-module by a coherent right X Y -module 0. Thanks to Lemma 6.3 the D | D | M epimorphism

0 0 X Y X S M ⊗D | D | −→ M −→ induces the epimorphism

( 0)b b 0 . Y X Y Xb Sb M ⊗D b| b D | −→ M −→ Hence, 1 charfb Sb ( b) φf− charXb Yb (( 0)bY ) | M ⊂ | M and the result will be true for f : X S Y S and the base change by b if it is true | −→ | for f : X Y Y Y and the base change by bY . In other words, we are reduced to the | −→ | obvious case where Y = S. 2

43 Theorem 6.5 Let f : X S Y S be a morphism of relative manifolds and let | −→ | b : Sb S be a base map. Denote by fb : Xb Sb Y b S b the image of f by the base −→ | −→ | change associated to b. Assume ( ,F ) is a good relative f-elliptic pair. Then M

a) ( b,Fb)isanfb-elliptic pair in some neighborhood of supp b, M M b) the canonical morphism

[f (F )] f ( F ) S! b b S ! b b | ⊗M −→ | b ⊗M is an isomorphism.

Proof: Since ( ,F ) is a relative elliptic pair, F is non characteristic for bX in a M neighborhood of supp and [12, Proposition 5.4.13] gives us an estimate of the micro- M support of Fb which together with the preceding proposition gives us (a). To prove part (b), we will use the graph factorization of f and part (c) of Propo- sition 6.2 to reduce the problem to the case where f : X S Y S is a relative | −→ | submersion. As in the preceding proposition, it is suﬃcient to treat the case Y = S. Assume

0 is a right -module and set = 0 . We have successively: X Y X Y X S M D | M M ⊗D | D |

[f ( )]b [f ( 0) Y S]b S! Y ! Y | M ' | M ⊗O D | [f ( 0)]b Y ! Y Y Yb Sb ' | M ⊗O b D | and

fb ( b) fb [( 0)bY X S ] Sb! Sb! X Y b b | M ' | M ⊗D b| b D | fb [( 0)b ] . Y ! Y Y Yb Sb ' | b M ⊗O b D | Hence, using Lemma 2.10, we see that the theorem will be true for f : X S Y S | −→ | and the base change by b if it is true for f : X Y Y Y and the base change by bY . | −→ | Finally, factorizing f and b through their graphs and using once more part (c) of Proposition 6.2, we see that it is suﬃcient to treat the case where f : Z S S S S × | −→ | is the second projection. We may also assume that the corresponding f-elliptic pair is b b op of the form ( ,G CI S)whereGand are objects of DIR c ( Z )andDgood( Z S S) M × M − D × | respectively, and that b : T S S is the ﬁrst projection. This product case is × −→ treated in Proposition 6.6 below. 2

Proposition 6.6 Let Z, S, T be complex analytic manifolds. Consider Z S as a × relative analytic manifold over S through the second projection .LetGbe an object b of DIR c ( Z ) and set F = G CI S . Consider the cartesian square − × Z S S × −→ p 2 b x η x Z T S T  S ×  × −→ ×

44 where the maps are the canonical projections. Assume that ( ,F ) is a relative elliptic M pair with -proper support on Z S S. Then the canonical map × | 1 L b− R (F Z S) 1 T S Z S S b− S ∗ ⊗M⊗D × | O × ⊗ O O × 1 1 L Rη (p− F p− 1 Z T S) p− Z S S −→ ∗ ⊗ M⊗ D × | O × × is an isomorphism. Proof: Thanks to the regularity theorem 2.15 and Lemma 3.3, it is equivalent to prove that the canonical morphism:

1 L b− R R om(D0F, IR Z S ) b 1 T S ∗ Z IR S S × − S × H M ⊗D × | F ⊗ O O 1 1 L Rη R om(p− D0F, p− IR p 1 Z T S T S ) ∗ − Z IR S S × × | × −→ H M ⊗ D × | F is an isomorphism. For short, let us denote by · (resp. · ) the source (resp. target) S1 S2 of the preceding arrow. Clearly, it is suﬃcient to prove that for any open polydisc ∆ of T , the induced morphism

Rb ( 1· ∆ S ) Rb ( 2· ∆ S ) (6.2) ∗ S | × −→ ∗ S | × is an isomorphism. We will compute this morphism explicitly as in the proof of Propo- sition 4.1. Using the notations introduced there, we already know that

L R R om(D0F, IR S Z S ) ∗ | Z IR S S × H M ⊗D × | F ∼ om(T · CI S, · Z S )= 2·. ∗ ZIR S S × −→ H × L ⊗D × | F R Since · has S-coherent cohomology, R2 O 1 L Rb ( 1· ∆ S ) Rb (b− 2· b 1 ∆ S ) ∗ S | × ' ∗ R ⊗ − OS O × L 2· b ∆S. 'R⊗OS ∗O× Moreover, we have:

L Rb ( 2· ∆ S ) R R om(D0F, IR p Z ∆ S ∆ S ) ∗ ∗ Z IR S S ∗ × × | × S | × ' H M ⊗D × | F om(T · CI S, · p Z ∆ S ∆ S ) . ZIR S S ' ∗ H × L ⊗D × | ∗ F × × | ×

Let us denote · this last complex. Since we have the isomorphism R4 1 ˆ ˆ p iW ∆ S iW− ∆ S Z ∆ S ∆ S Γ(W ; W ) Γ(∆; ∆) S ∗ ∗ × × ∗ × × F × × | × ' F ⊗ O ⊗O for any open subset W of Z, a direct computation shows that ˆ 4· ∼ 2· b ∆ S . R −→ R ⊗O S ∗ O × Clearly, the morphism (6.2) corresponds to the canonical morphism

L · b · ˆ b . 2 ∆ S 2 S ∆ S R ⊗OS ∗O × −→ R ⊗O ∗ O × Since · has S-coherent cohomology, Proposition 3.13 shows that it is an isomorphism R2 O and the conclusion follows. 2

45 6.2 K¨unneth formula

Theorem 6.7 Let f1 : X1 S Y 1 S and f2 : X2 S Y 2 S be two morphisms of | −→ | | −→ | relative manifolds. Assume

i) ( 1,F1) is a good relative f1-elliptic pair with f1-proper support, M

ii) ( 2,F2) is a good relative f2-elliptic pair with f2-proper support. M Then:

a) ( 1 2,F1 S F2) is a good relative f1 S f2-elliptic pair with f1 S f2-proper M ×S M × × × support,

b) the natural morphism

f (F ) f (F ) f f [(F F ) ( )] 1 S! 1 1 S 2 S! 2 2 1 S 2 S ! 1 S 2 1 S 2 | ⊗M × | ⊗M −→ × | × ⊗ M × M is an isomorphism.

Proof: Part (a) being obvious, we skip directly to part (b). Since

f (F ) 1 S! 1 1 | ⊗M has Y1 S -coherent cohomology, the formula D |

f1 S f2 =(idX S f2) (f1 S idX ) × 1 × ◦ × 2 allows us to restrict to the case f2 =idX2. So, we need only to prove that the canonical map

f (F ) (F ) f id (F F ) ( ) 1 S! 1 1 S 2 2 1 S X2 S! 1 S 2 1 S 2 | ⊗M × ⊗M −→ × | × ⊗ M × M is an isomorphism. Using the projection formula, we may get rid of F2. So we assume F2 =CIX . The problem being local on Y1 S X2, we may further assume that 2 is 2 × M equal to X2 S. D | The image of f1 : X1 S Y 1 S under the base change associated with 2 : X2 | −→ | −→ S is

f1 S idX : X1 S X2 X2 Y 1 S X 2 X 2 . × 2 × | −→ × | Hence, by Theorem 6.5, we have the isomorphism:

[f1 (F1 1)]2 ∼ ( f 1 S idX2 ) [(F1 1)2]. S! X2! | ⊗M −→ × | ⊗M By scalar extension, we get the isomorphism:

f (F ) f id [(F CI ) ( )] 1 S! 1 1 S X2 S ∼ 1 S X2 S! 1 S X2 1 S X2 S | ⊗M × D | −→ × | × ⊗ M × D | and the conclusion follows. 2

46 7 Microlocalization

Here, we shall prove that direct image commutes with microlocalization. More precisely, denote by X the sheaf of (ﬁnite order) microdiﬀerential operators on T ∗X (see [18] E or [19] for a detailed exposition). Consider a morphism f : X Y of complex analytic manifolds and the associated −→

diagram: w T ∗XXu t YT∗Y T∗Y f0× fπ and recall that the microlocal proper direct image of a right X -module is deﬁned E M through the formula

f ( )=Rf (tf 1 L ), ! π! 0− tf 1 X Y M M⊗ 0− EX E → where X Y denotes the micro-differential transfer module associated to f. E → Also recall that the microlocalization of a right X -module is the right X - D M E module defined on T ∗X by setting ME 1 = π− 1 . X π− X ME M⊗ X DX E In this section, we prove that, under the hypothesis of the finiteness theorem, we have [f ( F)] f[( F) ]. ! M⊗ E' ! M⊗ E This result was established by Kashiwara [9] when F =CIX and f is projective. It was also announced in a non proper case in [8].

7.1 The topology of the sheaf Y X(0) C | Let us show that the sheaf Y X(0) of [18] is naturally a sheaf of topological vector C | spaces and that its sections on a compact subset of TY∗ X form a DFN space. Proposition 7.1 Let X be a complex analytic manifold. Assume Y is a complex submanifold of X and denote by Y X (0) the sheaf of holomorphic microfunctions of C | order 0 on T ∗ X. Then, for any compact subset K T ∗ X,thespace Y ⊂ Y Γ(K; Y X (0)) C | has a canonical DFN topology.

Proof: Locally, we may use a coordinate system (x1,...,xd,y1,...,yn d)whereY is − deﬁned by the equations

x1 =0, ,xd =0. ··· Denote by (y1, ,yn d,ξ1,...,ξd) the corresponding coordinates on TY∗ X. It follows ··· − from [18, Theorem 1.4.5] that, for any open subset U of TY∗ X, the formula 0 (j) δ(p x, ξ )u(x, y)dx = aj(y,ξ)δ (p) (7.1) Z −h i j= X−∞

47 establishes a one to one correspondence between holomorphic microfunctions

u(x, y) Γ(U; Y X (0)) ∈ C | and sequences of homogeneous holomorphic functions

aj(x, ξ) Γ(U; T X(j)) (j 0) ∈ O Y∗ ≤ such that for any compact subset K U ⊂ 0 j − aj(x, ξ) K < + j= | | ( j)! ∞ X−∞ − for some >0. Let us ﬁrst construct the requested DFN topology in two special cases.

Case a. Assume K is a convex compact subset of T ∗ X on which ξk =0.Denoteby Y 6 p:T˙∗X P ∗ X the canonical projection. The preceding discussion shows that the Y −→ Y map

Γ(K; Y X (0)) Γ(p(K) 0 ; P X CI) C | −→ ×{ } O Y∗ × + j ∞ τ u(x, y) fk(y,ξ,τ)= a j(y,ξ/ξk) − 7→ j=0 j! X is an isomorphism. Using this isomorphism, we endow Γ(K; Y X (0)) with the usual C | DFN topology of Γ(p(K) 0 ; P X CI). If, moreover, ξ` =0onK, one has ×{ } O Y∗ × 6

fk(y,ξ,τ)=f`(y,ξ,τξk/ξ`).

Hence, the DFN topology of Γ(K; Y X(0)) does not depend on k. C | Case b. Let π denote the canonical projection of the bundle TY∗ X on its base Y identiﬁed to the zero section. Assume K is a convex compact subset of TY∗ X such that π(K) K. It follows from (7.1) that ⊂

Γ(K; Y X (0)) Γ(π(K); Y ) C | −→ O u(x, y) a0(y,0) 7→ is an isomorphism. We use this isomorphism to transport on Γ(K; Y X(0)) the usual C | DFN topology of Γ(π(K); Y ). O One checks easily that, if K1 K2 are two compact subsets of T ∗ X of the kind ⊂ Y treated in case (a) or (b) above, then the restriction map

Γ(K2; Y X (0)) Γ(K1; Y X (0)) C | −→ C | is continuous.

48 Let K be an arbitrary compact subset of TY∗ X. The preceding discussion shows that we can find a finite covering (Ki)i I of K by compact subsets such that Γ(Ki; Y X (0)) ∈ C | and Γ(Ki Kj ; Y X(0)) are DFN spaces. Thanks to the exact sequence ∩ C | α β 0 Γ(K; Y X (0)) Γ(Ki; Y X (0)) Γ(Ki Kj ; Y X (0)), | | | −→ C −→ i I C −→ i,j I ∩ C Y∈ Y∈ we may use α to transport on Γ(K; Y X(0)) the DFN topology of ker β.Toshow C | that this topology is independent of the chosen covering, it is sufficient to show that it is equivalent to the topology induced by a finer covering. Since such a topology is obviously weaker, the conclusion follows from the closed graph theorem. Since a direct computation shows that the above defined topology is independent of the chosen coordinate systems, the conclusion follows easily. 2

Corollary 7.2 Let X be a complex analytic manifold. Assume K is a compact subset of T ∗X.Then Γ(K; X (0)) E has a canonical DFN topology.

Proof: Apply the preceding proposition to ∆ X X (0). 2 C X | × Proposition 7.3 Let X, Z be complex analytic manifolds and let Y be a complex submanifold of X.WeidentifyT(∗ZY)(Z X)and Z TY∗ X.Wedenotebyq: × × × Z T∗X T ∗ X the second projection. Then, for any Stein compact subset K Z, ×Y −→ Y ⊂ one has

Rq![( Z Y Z X (0))K T X] Γ(K; Z ) ˆ Y X. C × | × × Y∗ ' O ⊗C |

Proof: Let S be a complex manifold. Denote by pS : Z S S the second projection. × −→ By classical results of analytic geometry, we know that

ˆ RpS ![( Z S)K S] Γ(K; Z ) S. O × × ' O ⊗O Using the explicit isomorphisms constructed in the proof of the preceding proposition, the conclusion follows easily. 2

Corollary 7.4 Let Z, Y be complex analytic manifolds and denote by

f : Z Y Y × −→ the second projection. Assume K is a Stein compact subset of Z.Then

Rfπ![( Z Y Y (0))K T Y ] Γ(K; Z ) ˆ Y(0). E × → × ∗ ' O ⊗E

Proof: Apply the preceding proposition to Z ∆ Z (Y Y )(0). 2 C × Y | × ×

49 7.2 Direct image and microlocalization Theorem 7.5 Assume f : X Y is a morphism of complex analytic manifolds and −→ ( ,F) is an f-elliptic pair on X with f-proper support. Then the canonical map M [f ( F)] f([ F] ) ! M⊗ E−→ ! M⊗ E b is an isomorphism in D ( Y ). coh E Proof: Recall that we have the commutative diagram

f 0 fπ w T ∗X u X Y T ∗Y T ∗Y ×

πX π πY

u u u w XXu Y ∼f Hence, we have successively

1 π− [f ( F)] 1 Y Y ! π− Y M⊗ ⊗ Y D E 1 L = Rf [π− ( F )] 1 π! X Y π− Y Y M⊗ ⊗DX D → ⊗ Y D E 1 L 1 = Rf [π− ( F ) 1 1 f− ] π! X Y fπ− π− Y π Y M⊗ ⊗DX D → ⊗ Y D E 1 L 1 1 = Rf [π− ( F) (π− 1 1 f− )]. π! π 1 X Y fπ− π− Y π Y M⊗ ⊗ − DX D → ⊗ Y D E Note that there is a canonical map

1 1 π 1 1 f . (7.2) − X Y f − π− π− Y X Y D → ⊗ π Y DY E −→ E → Hence, we get a canonical morphism

1 1 π− [f ( F)] 1 Y f [ π − ( F) 1 X]. (7.3) Y ! π− Y ! X π− X M⊗ ⊗ Y D E −→ M⊗ ⊗ X D E When f is a closed embedding, (7.2) is an isomorphism. Hence (7.3) is an isomor- b b phism for any Dcoh( X )andanyF DIR c ( X ). M∈ D ∈ − In the general case, consider the graph embedding

i : X X Y −→ × and the projection p : X Y Y. × −→ Since ( ,F )isanf-elliptic pair, the pair (i ,F CI Y )isp-elliptic. Since our result M !M × holds for closed embeddings and

i (F CIY) i( F), !M⊗ × ' ! M⊗ we are reduced to prove the theorem for the pair (i ,F CI Y )andthemapp. !M ×

50 We may thus assume that f is the second projection from X = Z Y to Y and b × that F = G CI Y where G is an object of DIR c ( Z ). Moreover, working as in 4, we × − § may also assume that = X where is a coherent -module. In this XY X Y M N⊗D | D N D | case,

1 π− [f ( F)] 1 Y Y ! π− Y M⊗ ⊗ Y D E 1 L 1 1 = Rf [π− ( F) (π− 1 1 f− )] π! π 1 X Y fπ− π− Y π Y M⊗ ⊗ − DX D → ⊗ Y D E 1 L 1 1 = Rfπ![π− ( (G CIY)) 1 (π− X 1 1 fπ− Y )] π− X Y fπ− πY− Y N⊗ × ⊗ D | O ⊗ O E and

1 1 f [πX− ( F) 1 X]=Rfπ![π− ( (G CIY)) 1 X Y ]. ! πX− X π− X Y M⊗ ⊗ D E N⊗ × ⊗ D | E → Hence, we are reduced to show that the canonical arrow

1 1 π− 1 1 f − (0) (0) X f − π− π Y X Y O ⊗ π Y OY E −→ E → induces an isomorphism

1 L 1 1 Rfπ![π− ( (G CIY)) 1 (π− X 1 1 fπ− Y (0))] (7.4) π− X Y fπ− πY− Y N⊗ × ⊗ D | O ⊗ O E 1 Rfπ [π− ( (G CIY)) 1 X Y (0)] ∼ ! π− X Y −→ N⊗ × ⊗ D | E → 1 As a matter of fact, (0) 1 f as a ( , )-bimodule and a X Y X Y f − (0) π− Y X Y Y E → 'E → ⊗ π EY E D | E scalar extension of (7.4) gives the theorem. Using the realiﬁcation process as in 4, we may assume from the beginning that Z § is a complexiﬁcation of a real analytic manifold M and that G is supported by M.

Since the result is local on T ∗Y (hence on Y ), we may assume also that has a N projective resolution · by ﬁnite free X Y -modules (see Proposition 3.1). L D | As for G, we may assume it is isomorphic to a bounded complex T · of the type

0 CIK CIK CIK 0 a,ia k,ik b,ib −→···ia ⊕Ia −→···ik ⊕Ik −→···ib ⊕Ib −→ ∈ ∈ ∈ where the sets Ik are ﬁnite and Kk,ik is a subanalytic compact subset of M (see Propo- sition 3.10). Hence,

(F CIY) · (T· CIY) N⊗ × 'L ⊗ × and the components of this last complex are ﬁnite direct sums of sheaves of the type

X Y CI K Y D | ⊗ × where K is a subanalytic compact subset of M. Note that

1 L 1 1 π− ( X Y CI K Y ) 1 (π− X 1 1 fπ− Y (0)) (7.5) π− X Y fπ− πY− Y D | ⊗ × ⊗ D | O ⊗ O E 1 1 π − ( ) 1 1 f − (0) ∼ X K Y f − π − π Y −→ O × ⊗ π Y O Y E 1 π− ( X Y CI K Y ) 1 X Y (0) (7.6) π− X Y D | ⊗ × ⊗ D | E → ∼ ( X Y (0))K T Y −→ E → × ∗

51 The right hand side of (7.5) is acyclic for fπ! thanks to usual properties of Stein compact subsets. Moreover, Corollary 7.4 shows that the right hand side of (7.6) is also b b acyclic for fπ . Hence, the morphism (7.4) of D ( Y (0)) is represented in C ( Y (0)) by ! E E the morphism 1 1 1 fπ![π− ( · (T · CI Y )) 1 (π− X 1 1 f − Y (0))] (7.7) π− X Y fπ− πY− Y π L ⊗ × ⊗ D | O ⊗ O E 1 f π [ π − ( · ( T · CI Y )) 1 X Y (0)] ! π− X Y −→ L ⊗ × ⊗ D | E → Let us denote by R· the complex

f![ · (T · CI Y ) X]. X Y L ⊗ ⊗ ⊗D | O Its components are direct sums of sheaves of the type

f![( X)K Y ] Γ(K; Z ) ˆ Y O × ' O ⊗O which are DFN-free Y -modules. It is easy to check that the Y -linear diﬀerential of O O R· is continuous with respect to the these natural topologies. Hence, we may consider R· as a topological complex of DFN-free Y -modules. Using Corollary 7.4, we have O successively

Rfπ![( X Y (0))K T Y ] Γ(K; Z ) ˆ Y(0) → × ∗ E ' O ⊗E 1 [Γ(K; ) ˆ π ] ˆ 1 (0) Z Y− Y π− Y ' O ⊗ O ⊗ Y OY E 1 π f [( ) ] ˆ 1 (0) Y− ! X K Y π− Y ' O × ⊗ Y OY E and (7.7) is represented as the canonical morphism 1 1 ˆ π− R· π 1 Y (0) π − R · π 1 Y (0). ⊗ − OY E −→ ⊗ − O Y E Since R· has Y -coherent cohomology, Proposition 3.13 allows us to conclude the proof. O 2

Corollary 7.6 Let be a coherent X -module endowed with a good filtration. As- M D sume: (i) f is proper on supp , M t 1 (ii) fπ is finite on f 0− (char ) (X Y T˙ ∗Y ),whereT˙∗Y =T∗Y T∗Y. M ∩ × \ Y Then, for j =0,Hj(f )is a flat connection (i.e. its characteristic variety is contained 6 !M in the zero section). Proof: The second hypothesis implies that f ( ) is concentrated in degree zero on ! ME T˙ ∗Y . The first hypothesis and Theorem 7.5 imply that (f ) f( ). !M E' ! ME j Hj[(f ) ] is contained in the zero section. Since is flat over Hence, for = 0, supp ! 1 6 M E E π− , the conclusion follows easily. 2 D This Corollary has important applications when studying correspondences of - D modules, such as, for example, the Penrose correspondence. We refer the interested reader to [5] for more details.

52 8 Main corollaries

8.1 Extension to the non proper case In this subsection, we shall generalize Theorems 4.2 and 5.15 to a non proper situation, using the techniques of [8, 12]. Let f : X S Y S be a morphism of complex manifolds over S and let ϕ : X | −→ | −→ IR be a real analytic function. Set

Λϕ = (x, dϕ(x)) : x X . { ∈ }

This is a Lagrangian submanifold of T ∗X (which is not conic for a non locally constant ϕ). We also associate to ϕ the following subsets of X:

Zt = x X : ϕ(x) t , { ∈ ≤ } Ut = x X : ϕ(x)

b op b Corollary 8.1 Let and F be objects of Dgood( X S) and DIR c ( X ) respectively M D | − and assume:

i) for each t IR , f is proper on supp supp F Zt, ∈ M∩ ∩ 1 ii) p− charf S( ) SS(F) TX∗ X, | M ∩ ⊂

iii) there is t0 IR such that ∈ 1 a 1 Λϕ (p− charf S( )+SS(F) ) π− (Zt0 ). ∩ | M ⊂ Then:

a) setting 1 Ft = jt j− F FU , ! t ' t the canonical morphisms:

f (F ) f ( F ) S! t S ! | ⊗M −→ | ⊗M f DF D f D F D S ( 0 t X S( )) S ( 0 X S ) | ∗ ⊗ | M ←− | ∗ ⊗ | M

are isomorphisms for t>t0,

b) both f (F ) and f (D F D ( )) S! S 0 X S | ⊗M | ∗ ⊗ | M b op are objects of Dgood( Y S), D |

53 c) the natural duality morphism:

f (D F D ( )) D f ( F ) S 0 X S Y S S ! | ∗ ⊗ | M −→ | | ⊗M is an isomorphism.

Note that replacing SS(F)bySS(F)a in hypothesis (iii), we get a similar conclusion after interchanging f and f . Also note that it would be possible to generalize to S! S a non proper situation| the results| ∗ of 6 but for the sake of brevity, we leave it to the § reader.

Proof: Let x X.Ifx supp supp F and x Zt ,thendϕ(x) char( )+ ∈ ∈ M∩ 6∈ 0 6∈ M SS(F)a by hypothesis (iii) and in particular dϕ(x) = 0. Applying Proposition 5.4.8 6 of [12], we ﬁnd for t>t0: + SS(Ft) SS(F)+IR Λϕ ⊂ + where IR Λϕ = (x; λdϕ(x)) : x X, λ > 0 . Since: { ∈ } 1 + 1 p− charf S( ) (SS(F)+IR Λϕ) TX∗X π− (Zt0), | M ∩ ⊂ ∪ again by hypothesis (iii), we obtain that ( , Ft) satisﬁes the hypothesis of Theorems 4.2 M and 5.15 for t>t0. Hence, the conclusions of these theorems apply to the pair ( , Ft) M and part (b) and (c) are consequences of part (a) which we shall now prove. First, we consider the morphism

f (F ) f ( F ). (8.1) S! t S ! | ⊗M −→ | ⊗M L Set G = F X S Y S. By Theorem 2.15, hypothesis (ii) and Proposition 5.4.14 X S ⊗M⊗D | D | → | of [12] we have: 1 SS(G) SS(F)+p− charf S( ). ⊂ | M Since 1 1 t p− charf S( )=p− charf S( )+ f0(X Y T∗Y), | M | M × the above morphism (8.1) is an isomorphism by Proposition 5.4.17 of [12]. To prove the second isomorphism in (a), consider the chain of isomorphisms which follows from the regularity theorem applied ﬁrst to Ft,thentoF:

L Rf (D0Ft DX S X S Y S) X S ∗ ⊗ | M⊗D | D | → | L Rf R om(Ft,DX S X S Y S) X S ' ∗ H | M⊗D | D | → | 1 L Rf Rjt j− R om(F, DX S X S Y S) t X S ' ∗ ∗ H | M⊗D | D | → | 1 L Rf Rjt j− (D0F DX S X S Y S). t X S ' ∗ ∗ ⊗ | M⊗D | D | → | Set L G = D0F DX S X S Y S. X S ⊗ | M⊗D | D | → |

54 Isomorphism (8.1) applied to (D0Ft, DX S ) tells us in particular that the projective | M system L Rf (D0Ft DX S X S Y S) X S ∗ ⊗ | M⊗D | D | → | 1 is essentially constant for t>t0. Hence, the projective system Rf Rjt jt− G is also ∗ ∗ essentially constant for t>t0 and using the Mittag-Leﬄer theorem we get the isomorphism 1 Rf G ∼ Rf Rjt jt− G ∗ −→ ∗ ∗ which completes the proof. 2

8.2 Special cases and examples In this subsection, we will consider various special situations and give the corresponding form of Theorem 4.2 and 5.15 leaving the reader do the same thing for Theorem 6.7. First, let us specialize our results to the non relative case taking S = pt . { } Corollary 8.2 Let f : X Y be a morphism of complex analytic manifolds. Assume −→ ( ,F ) is a good f-elliptic pair with f-proper support i.e.: M is an object of Db ( op), •M good DX b F is an object of DIR c ( X ) , • −

charf ( ) SS(F) T ∗ X, • M ∩ ⊂ X supp( ) supp(F ) is f-proper. • M ∩ Then

f ( F)is an object of Db ( op), • ! M⊗ good DY

f [D ( ) D0F ] D [ f ( F)]. • ! X M ⊗ −→∼ Y ! M⊗

When we take F =CIX in the preceding corollary we recover the coherence theorem for -modules of Kashiwara [9] (who treated only projective morphisms). Moreover, D using Corollary 8.1, we also recover the ﬁniteness theorem for non proper morphisms of [8] and the corresponding duality result of [24].

It is well known that an X-module is coherent if and only if the induced X - O F D module X is itself coherent. Moreover, this scalar extension process is com- F⊗OX D patible with direct images and duality (see Proposition 5.11). Applying the preceding corollary to the pair ( X,CI X ) we recover Grauert’s coherence theorem [6] and F⊗OX D Ramis-Ruget-Verdier’s relative duality theorem [15, 16] in the important special case of analytic manifolds. Taking Y = pt in the preceding corollary, we get the following absolute result: { } Corollary 8.3 Let X be a complex analytic manifold. Assume ( ,F ) is a good elliptic M pair with compact support i.e.:

55 is an object of Db ( op), •M good DX b F is an object of DIR c ( X ) , • −

char( ) SS(F) T ∗ X, • M ∩ ⊂ X supp( ) supp(F ) is compact. • M ∩ Then the complexes RΓ(X; F L ) and RΓ(X; R om ( F, Ω [d ])) X X X X M⊗ ⊗DX O H D M⊗ have ﬁnite dimensional cohomology and are dual one to each other.

In the special case where F =CIX, we get an absolute ﬁniteness and duality result for good X -modules which was considered by Mebkhout in [14]. For coherent analytic D sheaves, the preceding corollary corresponds to the very classical Cartan-Serre [4] and Serre [26]’s theorems. In the case Y = S, Theorem 4.2 and 5.15 give information on analytic families of absolute elliptic pairs.

Corollary 8.4 Let X S be a relative analytic manifold and let ( , F ) be a relative | M elliptic pair on X S i.e.: | b op is an object of Dgood( X S), •M D | b F is an object of DIR c ( X ) , • − 1 p − charX S( ) SS(F) TX∗ X, • | M ∩ ⊂ supp( ) supp(F ) is X-proper, • M ∩ where p : T ∗X T ∗ X S is the canonical projection. Then −→ | L Rf!(F X) and Rf!R om (F ,ΩX S[dX S]) X S X S ⊗M⊗D | O H D | ⊗M | | b are objects of D ( S), dual one to each other, i.e. the canonical morphism: coh O L Rf!R om (F ,ΩX S[dX S]) R om (Rf!(F X), S) X S S X S H D | ⊗M | | −→ H O ⊗M⊗D | O O is an isomorphism. Combining the preceding corollary with the base change formula, we get:

Corollary 8.5 Let X S be a relative analytic manifold. For any s S,denotebybs | ∈ the canonical inclusion of s in S. Assume ( ,F ) is a good relative elliptic pair with { } M -proper support on X S. Then for any s S,( b ,Fb ) is a good elliptic pair with | ∈ Ms s compact support (on a neighborhood of supp b in Xb , the ﬁber of X over s)and M s s the Euler-Poincar´e index:

L χ(RΓ(Xbs ; bs Fbs Xbs )) M ⊗ ⊗DXbs O is a locally constant function on S.

56 Proof: Let us denote by the ideal of holomorphic functions vanishing at s. The base I change Theorem 6.5 tells us that:

L L L RΓ(Xbs ; bs Fbs Xb )=[ S/ R ( F X)]s. X s S X S M ⊗ ⊗D bs O O I⊗O ∗ M⊗ ⊗D | O We know by the ﬁniteness theorem that

L R ( F X) X S ∗ M⊗ ⊗D | O has S coherent cohomology. Hence, it is locally quasi-isomorphic to a bounded com- O plex of ﬁnite free S-modules. The conclusion follows easily since [ S/ ]s =CI. O O I 2

Remark 8.6 Let P0 : E F , P 1 : E F be two complex analytic linear diﬀeren- −→ −→ tial operators between holomorphic vector bundles on X. Assume that their principal symbols induce the same morphism of ﬁber bundles

1 1 σ : π− E π − F. −→

Then, Pλ =(1 λ)P0+λP1 is a one parameter analytic family of operators with − principal symbols equal to σ. Combining this remark with the preceding corollary, we recover, for example, the fact that the index of an elliptic operator on a compact real analytic manifold depends only on its principal symbol. Let us now consider a few explicit examples. For the sake of brevity, we only consider non-relative situations.

Example 8.7 Let M be a real analytic manifold with X as a complexiﬁcation and M a good X -module. Then, as we have already noticed in the introduction, is elliptic D M on M in the classical sense if and only if ( , CI M ) is elliptic. In fact, SS(CI M )=T∗ X. M M Since CI M X = M, the sheaf of real analytic functions on M and ⊗O A

R om(D0CI M , X)= M H O B the sheaf of Sato’s hyperfunctions, the regularity theorem 2.15 entails the isomorphism:

R om ( , M ) R om ( , M ). (8.2) H DX M A ' H DX M B This is the Petrowski theorem for -modules which is often proved using micro-diﬀe- D rential equations as in [18]. Moreover, if M is compact and is good, Corollary 8.3 M asserts that the spaces

Hj(RΓ(M; R om ( , ))) = Extj (M; , ) M M M H D M B DM M B and n j L M H − (RΓ(M;ΩM )) = TorD (M;ΩM, ), M j n ⊗D M − M are ﬁnite dimensional and dual to each other. Note that for solutions of elliptic operators the duality and ﬁniteness theorems are well-known results.

57 Example 8.8 Let X be a complex manifold, U an open subset with real analytic boundary. Then ( , CI U ) is an elliptic pair if and only if the boundary ∂U is non M characteristic for ,thatis,char( ) T∗ X T∗ X. The regularity theorem yields M M ∩ ∂U ⊂ X the isomorphism:

R om ( , ( X)U ) ∼ R om ( , RΓU ( X)). (8.3) H DX M O −→ H DX M O In other words, the holomorphic solutions on U of the system extend holomorphically M through the boundary. If U is relatively compact, and is good, we get that the spaces j X M Ext (U, , X)andTorD (U;ΩX, ) are ﬁnite dimensional and dual to each other. X M O j n M NoteD that the regularity− theorem is due to Zerner [27] (for the 0-th cohomology) and [2], both in case of one equation with one unknown, then to Kashiwara [10] for systems. The ﬁniteness theorem is due to [2], this last result being extended in various directions by Kawai [13].

Example 8.9 One can generalize both preceding examples as follows. Let M be a real analytic manifold, X being a complexiﬁcation of M and let U be an open subset of

M with real analytic boundary. Then ( , CI U ) is an elliptic pair if and only if is M M elliptic on M on a neighborhood of U and moreover the conormal vectors to ∂U in M are hyperbolic with respect to . Then we get the isomorphism: M

R om ( , ( M )U ) ∼ R om ( , ΓU ( M )). H DX M A −→ H DX M B (i.e.: the hyperfunction solutions of on U are real analytic and extend analytically M through the boundary), and we also get ﬁniteness and duality results that we do not develop here.

Example 8.10 A general situation including the preceding examples is the following.

Let X = α Xα be a subanalytic µ-stratiﬁcation (cf. [12, Chap. VIII]) and assume: F SS(F) T ∗ X, ⊂ α Xα (8.4) ( char( ) T ∗ X T ∗ X α. M ∩ Xα ⊂ FX ∀

(In other words, F is locally constant on the strata Xα and these strata are non characteristic for .) M Then of course, the pair ( ,F) is elliptic. If, moreover, supp( ) supp(F )is M M ∩ compact we may apply Theorem 4.2 and Theorem 5.15 and we obtain new ﬁniteness and duality results.

b Example 8.11 For any F Ob(DIR c ( X )), the pair ( X,F) is elliptic. Since F L ∈ − O ' ΩX X F [ n]andD0F R om (F X, X), one recovers the classical ⊗ X O ⊗ − ' H DX ⊗O O ﬁnitenessD and duality theorem on constructible sheaves. In fact if M is a real analytic b manifold and i : M, Xdenote a complexiﬁcation of M,toG Ob(DIR c ( M )) one → ∈ − associates the elliptic pair ( X,i G). O ∗

58 Example 8.12 Let be a holonomic X -module and let x0 X.LetB(x0,ε)denote M D ∈ the open ball with center x0 and radius ε>0 in some local chart at x0.Byaresultof n Kashiwara [10], the pair ( , CI B(x0,ε)) is elliptic for 0 <ε 1. If X is open in CI and b M F Ob(DIR c ( X )) has compact support, one proves similarly that ( ,F CIB(0,ε))is ∈ − M ∗ elliptic for 0 <ε 1. (Here “ ” denotes the convolution of sheaves; cf. [12] Exercise ∗ 2.20.)

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