Regularity for elliptic pairs over <span class="mathjax-formula">$\mathbb {C}[[\hslash ]]$</span>

Regularity for elliptic pairs over C[[ h]]

DAVIDRAIMUNDO(*)

ABSTRACT - We extend the results of Schapira and Schneiders [12] on relative regularity, finiteness and duality (in the smooth case) of elliptic pairs to the framework ofD[[h]]-modules and constructible sheaves ofC[[h]]-modules.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 35A27, 32C38, 46L65.

KEYWORDS. Modules over the ring of formal differential operators, Elliptic pairs, Regularity, Finiteness and duality properties.

Introduction

The notion of elliptic pair goes back to [12], where the authors consider a morphism of complex analytic manifolds f :X!Y and say that a co- herentD_X-moduleMand anR-constructible sheaf ofC-modulesFform anf-elliptic pair if thef-characteristic variety ofMand the micro-support ofFdo not intersect outside the zero-section of the cotangent bundleTX.

Consider the constant mapa_X :X! fptg. IfXis the complexification of a real analytic manifold M and M is an elliptic system on M, then (M;C_M) is an a_X-elliptic pair. Elliptic pairs can thus be regarded as a generalization of elliptic systems. The functorial properties of elliptic pairs are studied in [12], where theorems of regularity, finiteness and duality are proved for these objects. As the authors point out, such theorems gen-

(*) Indirizzo dell'A.: Centro de MatemaÂtica e AplicacËoÄes Fundamentais da Universidade de Lisboa, Departamento de MatemaÂtica, FCUL, EdifõÂcio C6, piso 2, Campo Grande 1749-16, Lisboa, Portugal.

E-mail: [email protected]

The research of the author was supported by FundacËaÄo para a CieÃncia e Tecnologia and by FundacËaÄo Calouste Gulbenkian (Programa EstõÂmulo aÁ Investi- gacËaÄo).

eralize several classical results of DX-modules theory, complex analytic geometry and elliptic systems theory.

The purpose of this paper is to extend the main results of [12] to the framework of modules over the ringD_X[[h]], the ring of differential op- erators with a formal parameterh. This ring appeared first in [7] as an example of an algebra of formal deformation. Therefore, the machinery developed in [7] to perform the study of deformation-quantization modules also apply to the study ofD[[h]]-modules. Such study has been performed in [1] and [9].

Denote byC[[h]] the ring of formal power series onC. Consider the right exact functor that maps each sheaf of C[[h]]-modules F to F=hF regarded as a sheaf of C-modules. Consider also its left derived functor denoted by gr_h.

In this work we introduce the notion of f-elliptic pair overC[[h]] in a natural way: if M is a coherent D_X[[h]]-module and F is an R-con- structible sheaf of C[[h]]-modules, then (M;F) is an f-elliptic pair over C[[h]] if and only if (gr_h(M);gr_h(F)) is an elliptic pair in the sense of [12].

This allow us to prove theorems for f-elliptic pairs over C[[h]] using properties of gr_hgiven by [7] to reduce the proofs to the theorems of [12].

Let us mention our main results. In Theorems 3.1 and 3.7 we prove regularity properties for f-elliptic pairs over C[[h]]. These regularity theorems generalize a classical regularity property of elliptic systems in the real analytic setting: the complex of real analytic solutions of an elliptic system is isomorphic to the complex of hyperfunctions solutions.

In Theorem 3.9 we use the regularity of f-elliptic pairs overC[[h]] to give a finiteness criteria. Denote by f_!;h the proper direct image in the framework of DX[[h]]-modules introduced in [9]. The statement of the theorem is the following: given an f-elliptic pair (M;F), such that f is proper when restricted to supp (M)\supp (F) and such thatM is good, then the cohomologies of the direct imagef_!;h(M ^L_DX[[h]]F) are coherent overDY[[h]].

In Theorem 3.16 we prove a duality result in the case of a smooth morphism. It states that the direct image and the duality functor for D_X[[h]]-modules commute when applied to f-elliptic pairs that satisfy the finiteness criteria. The reason why we must restrict to the smooth case is that two fundamental properties hold in this case: the transfer module is coherent over DX[[h]] and the extension rings f^ÿ1(DX[[h]]) and (f^ÿ1DX)[[h]] are isomorphic. These properties are necessary to our con- struction of the duality morphism. We note that the smooth case includes the interesting case f ˆa_X.

In the last part of the paper we illustrate our results in some particular cases. For example, in the case where Xis the complexification of a real analytic manifoldM, we obtain regularity, finiteness and duality proper- ties on the sheaves of formal analytic functions and formal hyperfunctions onM.

1. Complements on formal extensions.

In view of our purpose it suffices to work on the complex analytic set- ting, although some results hold in a more general situation. In the sequel Xdenotes a complex analytic manifold of dimensiondX.

We follow the notations of [5]. Namely, ifRis a sheaf of rings onX, we denote by Mod(R) the category of left R-modules and by D^b(R) the bounded derived category of Mod(R). If R is coherent, we denote by Mod_coh(R) the full abelian subcategory of Mod(R) of coherent objects and byD^b_coh(R) the full triangulated subcategory of D^b(R) of objects with co- herent cohomology groups. In the sequel, when the base ring isCwe may omit it.

Let R₀ denote a sheaf ofC_X-algebras onX and set R:ˆR₀[[h]]ˆ P_n0R₀hⁿ. ThenRis a sheaf of algebras over the ringC[[h]], the ring of formal power series with complex coefficients. One uses the abbreviation C^h:ˆC[[h]].

Consider the following left exact functor studied in [1]:

()^h:Mod(R0) ! Mod(R)

N ! N ^h:ˆN [[h]]ˆlim

ƒn0(N R0R=h^n‡1R):

(1:1)

Recall that the sections of N ^hon an open subset ofXare formal power series of sections ofN on the same open subset.

One denotes by ()^Rh:D^b(R0)!D^b(R) the right derived functor of ()^h. For each F2D^b(R0), F^Rh is called the formal extension of F.

One says thatF2Mod(R0) ish-acyclic ifF^Rh'F^h.

PROPOSITION1.1 ([1], Proposition 2.5). Let N 2Mod(R₀) and sup- pose thatBis either a basis of open subsets of X, or a basis of compact subsets, such that H^j(S;N )ˆ0for all j>0and all S2 B. Then,N is

h-acyclic.

From now on assume thatR₀is anh-acyclicC_X-algebra.

LEMMA1.2. Let f :X!Y be a morphism of complex manifolds and assume that(f^ÿ1R0)^h'f^ÿ1R. Then, for eachN 2D^b(f^ÿ1R0), we have a canonical morphism inD^b(R):

Rf!(N ^Rh)!Rf!(N )^Rh:

PROOF. Using Proposition 2.1 of [1] and formula (2.6.26) of [5] we have inD^b(R):

Rf_!(N ^Rh) 'Rf_!RHom_fÿ1R0(f^ÿ1(R^loc=hR);N )

!RHom_R0(Rff^ÿ1(R^loc=hR);Rf_!N )

!RHomR0(R^loc=hR;Rf_!N )'(Rf_!N )^Rh:

p Consider also the right exact functor that maps eachM2Mod(R) into M=hM 2Mod(R0). We shall use its left derived functor

gr_h:D^b(R)!D^b(R₀); M!gr_h(M):ˆM^L_RR₀

which was studied in detail in [7]. Recall that gr_hcommutes with tensor products, with RHomand also with direct images, proper direct images and inverse images of sheaves. Recall also that M2Mod(R) has no h- torsion if gr_h(M) is concentrated in degree 0, that is, if h:M!M is injective.

One says that M2Mod(R) is h-complete if the canonical morphism M !lim

n

M=h^n‡1Mis an isomorphism. SetR^loc:ˆC((h))_XC^h

XR, where C((h)) denotes the field of Laurent series with complex coefficients. One says thatM2D(R) is cohomologically complete if RHom_R(R^loc;M)ˆ0. The notions ofh-complete object and cohomologically complete object don't de- pend on the base ring.

We refer [7] for a comprehensive study of cohomologically complete objects. Let us just mention some facts that we shall use in the sequel: gr_h is conservative when restricted to the triangulated subcategory ofD^b(R) consisting of cohomologically complete objects (cf. Corollary 1.5.9 of [7]) andN ^Rh is cohomologically complete for every N 2D^b(R₀) (cf. Propo- sition 1.2 of [1]).

In view of Proposition 1.5 below let us fix a morphism of complex analytic manifolds f :X!Y and let R₀ be an h-acyclic C^h-algebra on Y.

REMARK1.3. There is a canonical morphism of sheaves ofC^h-algebras f^ÿ1Rˆf^ÿ1R^h₀ !(f^ÿ1R0)^hinduced by the morphisms

f^ÿ1R^h₀ !f^ÿ1(R0C^h_Y=h^n‡1C^h_Y)'f^ÿ1R0C^h_X=h^n‡1C^h_X;

and by the universal property of projective limits. Hence, there is a ca- nonical functor F2Mod((f^ÿ1R0)^h)7!F2Mod(f^ÿ1(R^h₀)).

ASSUMPTION1.4. There exists a basisBeither of open subsets of Y or of compact subsets of Y such that H^j(S;R₀)ˆ0, for all j>0and for all S2 B.

PROPOSITION 1.5. Suppose that f :X!Y is smooth and that R0

verifies Assumption1.4.Then f^ÿ1Rÿ! (f^ÿ1R0)^h.

PROOF. Consider the canonical morphism from Remark 1.3. Note that the corresponding canonical morphism gr_h(f^ÿ1R)!gr_h((f^ÿ1R₀)^h) is an isomorphism, since one has gr_h(f^ÿ1R)'gr_h((f^ÿ1R₀)^h)'f^ÿ1R₀. Hence, in view of [7, Cor. 1.5.9], it suffices to show that bothf^ÿ1Rand (f^ÿ1R0)^hare cohomologically complete objects.

The ringR'R^R₀^his cohomologically complete. Sincefis smooth,Ris non-characteristic for f andf^ÿ1Ris cohomologically complete by Propo- sition 2.15 of [9].

We shall use Proposition 1.1 to prove that (f^ÿ1R₀)^Rhis concentrated in degree 0, thus (f^ÿ1R₀)^his cohomologically complete.

The result is now checked locally, so we can assume thatXˆX⁰Y and that f :X!Y is the canonical projection.

Let us consider the case whereBis a basis of open subsets ofY. It is enough to show that there exists a basisB⁰of open subsets ofX⁰Y such thatH^j(S⁰;f^ÿ1R₀)ˆ0 for allj>0 and allS⁰2 B⁰.

Consider the basisB⁰formed by the open setsV⁰VX⁰Y such that V⁰ is an open ball ofX⁰ (hence, contractible) andV 2 B. We are in the conditions to apply [5, Proposition 2.7.8]. Hence, for any j>0 one has:

H^j(V⁰V;f^ÿ1R0) ' H^j(f_j^ÿ1_V0V(V);f_j^ÿ1_V0VR0j_V)'H^j(V;R0j_V)ˆ0:

The case whereBis a basis of compact subsets ofXis similar, taking

closed balls onX⁰instead of open balls. p

R-constructible sheaves ofC[[h]]-modules.

As usual denote by SS(F) the micro-support of an objectF2D^b(C^h_X), a closed involutive subset of the cotangent bundle TX. One has the esti- mative SS(gr_h(F))SS(F). The equality SS(gr_h(F))ˆSS(F) holds ifFis cohomologically complete (cf. Proposition 1.15 of [1]), then in such case one also has supp (F)ˆsupp (gr_h(F)).

LetKbe eitherCorC^h. One denotes byD^b_Rÿc(K_X) the bounded derived category of sheaves ofK-modules withR-constructible cohomology.

Consider the constant mapa_X :X! fptgand denote byv^K_Xthe dualizing sheaf in the categoryD^b(K_X). Recall that one hasv^K_X 'a^!_XK'K_X[2d_X].

We shall use the duality functors below:

D⁰_KX :D^b(KX)!D^b(KX); F7!RHom[KX](F;KX);

DKX :D^b(KX)!D^b(KX); F7!RHom[KX](F;v^K_X):

They induce functors D⁰_KX;DKX :D^b_Rÿc(KX)!D^b_Rÿc(KX) which satisfy the microlocal relation: SS(D⁰_KXF)ˆSS(DKXF)ˆSS(F)^a, whereadenotes the opposite map onTX.

Any F2D^b_Rÿc(C^h_X) is cohomologically complete by Proposition 1.6 of [1]. Hence, gr_h:D^b_Rÿc(C^h_X)!D^b(C_X) is conservative and preserves the micro-support. It is also known thatR-constructible sheaves areh-acyclic (Corollary 2.6 of [1]).

The next proposition shows that F^h_C^h

X is an exact functor in Mod(C^h_X) for eachF2Mod_Rÿc(C_X).

PROPOSITION1.6. Let F2D^b_Rÿc(CX). Then we have F^Rh'FC^h_X. PROOF. First we remark that by replacingF with an almost free re- solution ofF (in the sense of the Appendix of [6]) one easily reduces the proof to the caseFˆCU,Ubeing an open subanalytic relatively compact subset ofX. This reduction uses the fact thatR-constructible sheaves areh- acyclic.

Therefore, it is enough to note that for each open subanalytic subsetU we have a canonical isomorphism:

CUC^h_X ÿ! C^h_U:

In fact, the stalks (C_UC^h_X)_xand (C^h_U)_xare both isomorphic toC^hifx2U,

and both vanish ifx2=U. p

As a consequence of Proposition 1.6 above and Lemma 1.9 of [1], we have:

COROLLARY 1.7. For F2D^b_Rÿc(C_X) and G2D^b(C_X), there are iso- morphisms inD^b(C^h_X):

(i) RHom_C^h

X(F^Rh;G^Rh)'(RHom(F;G))^Rh (ii) F^RhL_Ch

XG^Rh'FG^Rh:

2. Elliptic pairs overC[[h]]

D[[h]]-modules.

LetO_X,V_XandD_Xdenote the ring of holomorphic functions onX, the sheaf of holomorphic forms of maximal degree onX, and the ring of linear differential holomorphic operators onX, respectively.

As usual, Modgd(DX) denotes the full subcategory of Modcoh(DX) con- sisting of goodDX-modules (in the sense of [4]) andD^b_gd(DX) denotes the full subcategory ofD^b_coh(D_X) consisting of objects with good cohomology.

An objectM2D^b_coh(D^h_X) is said to be good if gr_h(M)2D^b_gd(D_X). De- note byD^b_gd(D^h_X) the full triangulated subcategory ofD^b_coh(D^h_X) consisting of good objects.

We shall use the duality functors below:

D_h;X :D^b((D^h_X)^op)!D^b((D^h_X)^op); M7!RHom_D^h

X(M;V_X[d_X]_OX D^h_X) D⁰_h;X :D^b(D^h_X)!D^b((D^h_X)^op); M7!RHom_Dh

X(M;D_X):

Denote also by D_X and D⁰_X their counterparts forD-modules.

The ringsD^h_XandD^h_X^opare algebras of formal deformation in the sense of [7] and the machinery developed in loc. cit. apply to the study ofD^h_X- modules. In particular, objects belonging toD^b_coh(D^h_X) are cohomologically complete and Theorem 1.6.4 of [7] provide a useful coherence criteria for D^h_X-modules. Recall also that coherent D_X-modules are h-acyclic (cf.

Corollary 2.6 of [1]).

In the sequel most of our results are stated for rightD^h_X-modules but one can get similar results for left D^h_X-modules. Indeed, the category of left D^h_X-modules and the category of right D^h_X-modules are equivalent.

Hence, we don't need to distinguish left and rightD^h-modules.

Direct images.

From now on, f :X!Y denotes a complex morphism between two complex analytic manifolds of dimensionsd_X andd_Y, respectively.

Consider the usual transfer-module D_X!Y :ˆO_Xfÿ1OYf^ÿ1D_Y with its structure of (DX;f^ÿ1DY)-bimodule. Denote byfandf_!the functors of direct image and proper direct image in theD-modules framework.

Set DX!Y;h:ˆlim

n

(OX_f^ÿ1_OY f^ÿ1(D^h_Y=h^n‡1D^h_Y)):

Since each component of the projective limit has a natural structure ofh- torsion (D^h_X;f^ÿ1D^h_Y)-bimodule, thenD_X!Y;his also a (D^h_X;f^ÿ1D^h_Y)-bimod- ule. Hereafter, when there is no risk of confusion, we use the following abbreviations:K_h:ˆD_X!Y;handK :ˆD_X!Y.

Note thatK^his also a (D^h_X;f^ÿ1D^h_Y)-bimodule (cf. Remark 1.3) and one has an isomorphism of bimodules Kh'K^h. Moreover, Kh'K^Rh is cohomologically complete, free ofh-torsion and gr_h(K_h)'K (Proposi- tion 5.5 of [9]).

The direct image functor and the proper direct image functor in the framework of D^h-modules are denoted respectively by f_;h and f_!;h and defined by:

f_;h :D^b(D^h_X)!D^b(D^h_Y); M7!Rf(M^L_Dh XK_h);

f_!;h :D^b(D^h_X)!D^b(D^h_Y); M7!Rf_!(M^L_Dh XK_h):

Direct images forD^h-modules are introduced in [9] (where the transfer module in theh-setting is simply denoted byK).

Clearly, forM2D^b(D^h_X) the isomorphisms gr_h(f_;h(M))'f(gr_h(M)) and gr_h(f_!;h(M))'f_!(gr_h(M)) hold.

f -characteristic variety.

Following [12], the f-characteristic variety ofM2D^b_coh(D_X), denoted by char_f(M), is a closed conic analytic subvariety ofTXdepending on f and satisfying the formula

SS(M ^L_DXK)char_f(M):

(2:1)

Note that charaX is the usual characteristic variety of M, denoted simply by char(M). In this case, the inclusion (2.1) gives the well-known estimative: SS(Sol(M))char(M). Here Sol denotes the solutions func-

tor in the D-module framework, whose counterpart in the D^h-modules framework is the functor Solhstudied in [1]:

Sol_h:D^b_coh(D^h_X)^op!D^b(C^h_X); M7!RHom_D^h

X(M;O^h_X):

DEFINITION2.1. The f-characteristic variety ofM 2D^b_coh(D^h_X) is de- noted by char_f;h(M) and defined by char_f;h(M):ˆchar_f(gr_h(M)).

For anyM2D^b_coh(D^h_X), char_aX;h(M) coincides with the characteristic variety ofMdenoted by charh(M).

LEMMA 2.2. For any M 2D^b_coh(D_X^h), we have SS(M^L_D^h

XK_h) char_f;h(M).

PROOF. Since M is coherent and Kh is cohomologically complete, M ^L_D^h

XKh is also cohomologically complete by Proposition 1.6.5 of [7].

Hence,

SS(M ^L_Dh

XK_h)ˆSS(gr_h(M)^L_DXK):

Therefore, the result follows from estimative (2.1). p We remark that SS(Solh(M))ˆcharh(M) is already proved in [1].

Elliptic pairs.

DEFINITION 2.3. A pair (M;F) with M2D^b_coh(D^h_X^op) and F2D^b_Rÿc(C^h_X) is anf-elliptic pair overC^hif char_f;h(M)\SS(F)T_XX. If in additionM 2D^b_gd(D^h_X^op), then (M;F) is said a goodf-elliptic pair over C^h. The support of the pair (M;F) is the intersection supp (M)\supp (F).

Since gr_hpreserves the micro-support ofR-constructible sheaves and the characteristic variety of coherentD^h-modules we have:

PROPOSITION2.4. A pair(M;F)is an f -elliptic pair overC^h if and only if(gr_h(M);gr_h(F))is an f -elliptic pair overC(in the sense of[12]).

If (M;F) is anf-elliptic pair overC^h, then (D_h;XM;D⁰_Ch

XF) is also anf- elliptic pair overC^h, the dual f-elliptic pair of (M;F).

Assume thatXis the complexification of a real analytic manifoldM.

DEFINITION2.5. We say that a coherentD^h_X-module M is an elliptic D^h_X-module if (M;C^h_M) is an aX-elliptic pair overC^h. We say that an op- eratorP2D^h_X is an elliptic operator ifD^h_X=D^h_XPis an ellipticD^h_X-module.

In other words,M2Mod_coh[D^h_X] is elliptic if char_h(M)\T_MXT_XX.

Moreover, one deduces from Lemma 3.5 of [1] thatP2D^h_Xis elliptic if and only if it is locally written as PˆP0‡hP⁰ for some P⁰2D^h_X and P0 an elliptic operator in the classical sense. Take a system of holomorphic co- ordinates (x;h) on TX. P0 is elliptic if its principal symbol satisfies s(P₀)((x;ih))6ˆ0 forh6ˆ0. Take, for example,XˆCⁿ,MˆRⁿand denote byDthe Laplace operator. Then,PˆD‡hP⁰is elliptic for anyP⁰2D^h_X.

The meaning of elliptic pairs on the real analytic setting illustrates why one can regard the theory of elliptic pairs (overC^h) as a natural general- ization of the theory of elliptic systems on real analytic manifolds.

3. Theorems on elliptic pairs overC[[h]]

Regularity theorem.

THEOREM3.1. Let(M;F)be an elliptic pair overC^h. Then, the natural morphism below is an isomorphism inD^b(C^h_X):

F^L_C^h

X(M ^L_D^h

XKh)!RHom_C^h

X(D⁰_Ch

XF;M^L_D^h

XKh) (3:1)

PROOF. The morphism (3.1) is induced by the isomorphism F'D⁰_Ch

XD⁰_Ch

XFand by the canonical morphism:

D⁰_Ch XD⁰_Ch

XF^L_C^h

X(M ^L_D^h

XKh)!RHom_C^h

X(D⁰_Ch

XF;M^L_D^h

XKh):

Note that D⁰_Ch

XFhasR-constructible cohomology and SS(D⁰_Ch

XF)ˆSS(F)^a. The transversality condition on the pair (M;F) together with Lemma 2.2 entail:

SS(M ^L_D^h

XKh)\SS(D⁰_Ch

XF)^aT_XX:

The conclusion follows by Proposition 5.4.14 of [5]. p REMARK3.2. The isomorphism (3.1) is written in the following form in terms of leftD^h_X-modules:

RHom_D^h

X(M;D⁰_ChF^L_C^h

XK_h)ÿ! RHom_D^h

X(M;RHom_C^h

X(F;K_h)):

We want to refine the regularity property in the case whereFis the formal extension of an object inD^b_Rÿc(CX). Let us start with some auxiliar results.

LEMMA 3.3. Let F;G2Mod(C_X). There is a natural morphism in Mod(C^h_X):

FG^h!(FG)^h: (3:2)

PROOF. It is enough to note that there is a projective system of morph- ismsFG^h!F(G(C^h=h^n‡1C^h))ÿ! (FG)(C^h=h^n‡1C^h). p LEMMA 3.4. Let F;G2D^b(C_X). Then there is a natural bifunctorial morphism inD^b(C^h_X):

FG^Rh!(FG)^Rh: (3:3)

PROOF. (i) First, fixF2Mod(CX) and consider the two functors from Mod(C_X) to Mod(C^h_X):u1:G7!FG^handu2:G7!(FG)^h. By Lemma 3.3, there is a morphism of functors u:u₁!u₂. Since both functors are left exact, this morphism u extends to the derived functors and we get the morphism (3.3) for a fixedF2ModRÿc(CX).

(ii) Now let us fix G2D^b(CX) and consider the two functors from Mod(C_X) toD^b(C^h_X):l1:F7!FG^Rh andl2:F7!(FG)^Rh. By (i) there exists a morphism of functorsv:l₁!l₂. Both functors extend naturally to the category of bounded complexes C^b(Mod(C_X)) and send complexes quasi-isomorphic to zero to objects isomorphic to zero in D^b(C^h_X). Hence, both functors extend toD^b(CX) as well as the morphism of functorsv. p LEMMA 3.5. For F;G2D^b(CX), we have a commutative diagram in D^b(C^h_X):

(3:4)

such that the morphism in the horizontal arrow is the one given by Lemma 3.4.

PROOF. The oblique arrow is the composition of two canonical morphisms:

RHom(F;C)G^Rh!RHom(F;G^Rh)ÿ! RHom(F;G)^Rh:

The vertical arrow is defined by applying the functor ()^Rhto the canonical morphism RHom(F;C)G!RHom(F;G). The commutativity of the

diagram is obvious. p

REMARK 3.6. Assuming that G2Mod(D^h_X) in Lemma 3.3, then the morphism (3.2) isD^h_X-linear. Similarly, ifG2D^b(D^h_X), then the morphism (3.3) is a morphism inD^b(D^h_X) and the diagram in Lemma 3.5 is a com- mutative diagram inD^b(D^h_X). In the sequel we shall use such diagram in the caseGˆK_h.

THEOREM3.7. Let F2D^b_Rÿc(C_X)and suppose that(M;F^h)is an f - elliptic pair overC^h. Then there is a commutative diagram of isomorph- isms inD^b(C^h_X):

(3:5)

PROOF. First note that we obtain the diagram (3.5) by applying the functor RHom_Dh

X(M;) to the commutative diagram provided by Lemma 3.5:

We also use the fact that D⁰_Ch XF^hL_C^h

XK_his isomorphic to D⁰FK_h. Note that the oblique arrow is an isomorphism since it is the compo- sition of two canonical isomorphisms:

RHomDX^h(M;D⁰_Ch XF^hL_C^h

XKh) ÿ! RHom_D^h

X(M;RHom_C^h

X(F^h;Kh)) ÿ! RHom_Dh

X(M;RHom(F;Kh)^Rh);

The first one results from applying the regularity theorem 3.1 to the f-el- liptic pair (M;F^h) (see also Remark 3.2) and the second isomorphism results from Lemma 1.7.

The vertical arrow is also an isomorphism inD^b(C^h_X). In fact, we con- clude from [7, Prop. 1.5.10] and [1, Prop. 2.2] that the objects on both sides of the vertical arrow are cohomologically complete. Hence, since gr_h is

conservative on cohomologically complete objects, it is enough to prove that the canonical morphism below is an isomorphism:

(3:6) RHomDX(gr_h(M);D⁰FK)!RHomDX(gr_h(M);RHom(F;K)):

In fact, (grh(M);F) is an f-elliptic pair overCand the morphism (2.2) is precisely the regularity isomorphism for elliptic pairs over C applied to (gr_h(M);F) (see Theorem 2.15 of [12]).

We have proved that the oblique and vertical arrows in diagram (3.5) are isomorphisms. The commutativity of the diagram allow us to conclude that the horizontal arrow is also an isomorphism. p

Using the functorial properties of ()^Rhwe also get:

COROLLARY3.8. Let(M;F)be an f -elliptic pair overC. Then,(M^h;F^h) is an f -elliptic pair over C^h and there are canonical isomorphisms in D^b(C^h_X):

RHom_Dh

X(M^h;D⁰_Ch XF^hL_Ch

XK_h) ' RHom_DX(M;D⁰FK)^Rh

' RHom_DX(M;RHom_CX(F;K))^Rh: Finiteness theorem.

THEOREM3.9. Let(M;F)be a good f -elliptic pair overC^hand sup- pose that f is proper when restricted to the support of (M;F). Then, f_!;h(M^L_Ch

XF)is an object ofD^b_gd(D^h_Y).

PROOF. Note that by Theorem 1.6.4 of [7] it suffices to prove that:

(i) f_!;h(F^L_C^h

XM) is cohomologically complete;

(ii) gr_h(f_!;h(F^L_C^h

XM)) is an object ofD^b_gd(D^h_Y).

Set L :ˆ(F^L_C^h

XM)^L_D^h

XKh. The regularity theorem 3.1 yields the isomorphism:

L 'RHom_Dh X(D⁰_Ch

XF;M ^L_Dh XK_h):

SinceM ^L_D^h

XK_his cohomologically complete,L is also cohomologically complete in view of Propositions 1.5.10 of [7]. Finally, f_!;h(F^L_Ch

XM)ˆ Rf!(L) is cohomologically complete by Proposition 1.5.12 of [7] and since Rf and Rf_!coincide by the hypothesis on the support.

On the other hand, note that (gr_h(M);gr_h(F)) is an f-elliptic pair over Cthat satisfies the conditions of the finiteness theorem for elliptic pairs over C. Hence, gr_h(f_!;h(F^L_Ch

XM))'f_!(gr_h(F)gr_h(M)) is an object of

D^b_gd(DY) by Theorem 4.2. of [12]. p

Duality theorem.

The results in this paragraph are obtained assuming thatf :X!Yis a smooth morphism of complex manifolds.

PROPOSITION3.10. If f :X!Y is smooth, then f^ÿ1D^h_Y '(f^ÿ1D_Y)^h. PROOF. This is a particular case of Proposition 1.5 choosing forBthe

family of compact Stein subsets ofY. p

The rings f^ÿ1D^h_Y and (f^ÿ1D_Y)^h are not isomorphic in general: if i:Y,!X is the embedding of a closed submanifold of dimensiond_Y5d_X, theni^ÿ1D^h_X !(i^ÿ1D_X)^his a monomorphism which is not an epimorphism.

Note that in the smooth caseKhis a coherent leftD^h_X-module. Hence, for any coherent rightD^h_X-moduleM, one has:

M_D^h

X K^h'M_D^h

X (D^h_XDX K)'MDX K: (3:7)

By the isomorphisms in (3.7), the object MDX K has a structure of f^ÿ1(D^h_Y^op)-module. Passing to the derived category and applying Rf_!, one concludes thatf_!(M) is an object ofD^b(D^h_Y^op) and that it is isomorphic to f_!;h(M) for anyM2D^b_coh(D^h_X^op). In other words, the direct image ofMin theh-setting coincides with its direct image as aD-module. We use this fact in the sequel.

LEMMA 3.11. Let M 2D^b_coh(D^op_X). There is a morphism f_!;h(M^h)! f_!(M)^RhinD^b(D^h_Y^op).

PROOF. First note that we have a chain of isomorphisms in D^b(f^ÿ1D^h_Y^op):

M^hL_DXK ' M^hL_DXD⁰_XD⁰_XK ' RHom_DX(D⁰_XK;M_h) ' RHom_DX(D⁰_XK;M)^Rh:

The first and second isomorphisms follow from the coherence ofK over DX. The third follows from the properties of ()^Rh(Lemma 2.3 of [1]).

Applying Rf_!we get the following isomorphism inD^b(D^op_Y):

f_!;h(M^h)'Rf!((M^LDXK)^Rh):

Finally, Lemma 1.2 entails the following morphism inD^b(D^h_Y^op):

f_!;h(M^h)!Rf!((M^LDXK))^Rhˆf_!(M)^Rh:

p IfMis a right (DX;DX)-bimodule, its direct image as a bimodule is the object in the derived categoryD^b(D^op_Y D^op_Y) defined by:

f!(M):ˆRf_!((M^LDXK)^LDXK):

In particularV_XOXD_Xis a right (D_X;D_X)-bimodule and one has the so- called differential trace morphism inD^b(D^op_Y D^op_Y):

trf :f

!(VX[dX]OXDX)!VY[dY]OYDY: (3:8)

Note that ifM is a right (D^h_X;D^h_X)-bimodule thenf

!;h(M):ˆf

!(M) is an object ofD^b(D^h_Y^op_ChD^h_Y^op).

PROPOSITION 3.12. The morphism f induces a differential trace morphism in the derived categoryD^b(D^h_Y^op_C^hD^h_Y^op):

tr_f;h:f

!;h(V_X[d_X]_OX D^h_X)!V_Y[d_Y]_OYD^h_Y:

PROOF. We have the following isomorphisms inD^b(D^h_Y^op_C^hD^h_Y^op):

f!;h(VX[dX]OXD^h_X) ' f_!;h(VX[dX]OXKh) ' f_!;h((VX[dX]OXK)^h)

The first one results from the associativity of tensor products. The second one results fromV_X[d_X]_OX K_h'(V_X[d_X]_OX K)^h.

The morphism trf;h is then constructed composing the morphisms below:

f_!;h((V_X[d_X]_OXK)^h)!f_!(V_X[d_X]_OXK)^Rh!V_Y[d_Y]_OYD^h_Y:

The first morphism is an application of Lemma 3.11 which, in this case, preserves the bimodule structures involved. The second morphism is the formal extension of the classical trace morphism (3.8). p REMARK3.13. Consider the absolute casef ˆa_X. In this case one gets K_h'O^h_X and the following isomorphisms hold inD^b(C^h):

aX!;h(VX[dX]OXD^h_X)'RGc(X;C^h_X[2dX])'RGc(X;v^h_X):

Thus, the trace morphism tr_aX;h coincides with the morphism RG_c(X;v^h_X)!C^hinduced by the natural transformation Ra_X!a^!_X !id.

LEMMA 3.14. Let M2D^b(D^h_X^op). We have a canonical morphism in D^b(D^h_Y^op):

f_!;h(D_h;XM)!D_h;Y(f_!;hM):

(3:9)

PROOF. First note that there is a basis change morphism inD^b(D^h_Y^op):

f_!;h(D_h;X(M))'Rf_!(RHom_Dh

X(M;V_X[d_X]_OXK_h))

!RHom_D^h

Y(Rf!(M ^LDXK);Rf!((VX[dX]OX Kh)^LDXK)) 'RHom_D^h

Y(f_!;h(M);f

!;h(VX[dX]OXD^h_X)):

We obtain (3.9) composing the above morphism with the morphism induced by Proposition 3.12 on the second term of the RHom. p COROLLARY 3.15. For M2D^b(D^h_X^op) and F2D^b(C^h_X), there is a canonical morphism inD^b(D^h_Y^op):

f_!;h(D⁰_Ch XF^L_Ch

XD_h;X(M))!D_h;Y(f_!;h(F^L_Ch XM)):

(3:10)

PROOF. Note that there is a canonical morphism inD^b(D^h_X^op):

D⁰_Ch XF^L_C^h

XD_h;X(M)!D_h;X(F^L_C^h

XM):

(3:11)

The morphism (3.10) is then obtained as a composition of morphisms:

f_!;h(D⁰_Ch XF^L_Ch

XD_h;X(M)) ! f_!;h(D_h;X(F^L_Dh XM))

! D_h;Y(f_!;h(F^L_Ch XM));

where the first arrow results from (3.11) and the second arrow results

from (3.9). p