Constructibility of tempered solutions of holonomic D-modules
aa r X i v : . [ m a t h . AG ] N ov Constructibility of tempered solutionsof holonomic D -modules Giovanni Morando
Abstract
In this paper we prove the constructibility on the subanalytic sitesof the sheaves of tempered holomorphic solutions of holonomic D -modules on complex analytic manifolds. Such a result solves a con-jecture of M. Kashiwara and P. Schapira ([19]). Contents
Introduction 11 Notations and review 3 D -modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Tempered De Rham complexes . . . . . . . . . . . . . . . . . 81.4 Asymptotic decomposition of meromorphic connections . . . . 13 R -sa-constructibility of the tempered De Rham complex ofholonomic D -modules 16 R Γ [ X \ S ( M )] M . . . . . . . . . . . . . . . . . . . . 172.2 The case of R Γ [ S ( M )] M . . . . . . . . . . . . . . . . . . . . . . 22 References 24
Introduction
The constructibility of the complexes of holomorphic solutions of holonomic D -modules on complex manifolds, proved by Masaki Kashiwara in [12], is Primary 32C38; Secondary 32B20 32S40 14Fxx.
Keywords and phrases: D -modules, irregular singularities, tempered holomorphic func-tions, subanalytic. Introduction a fundamental result in the algebraic study of systems of partial differen-tial equations. In its general form, it states that if M , N are holonomic D -modules on a complex manifold X , the cohomology groups of the com-plex R H om D X ( M , N ) are locally constant sheaves with finite dimensionalstalks when restricted to the strata of some complex analytic stratification.In particular, this implies (for N ≃ O X ) that the complex of holomorphicsolutions of a holonomic D -module on a complex manifold is constructible.A part from its intrinsic importance, this results is at the base of the state-ment of the Riemann–Hilbert correspondence, one of the most importantachievements in D -module theory. Such a correspondence states that thederived functor of holomorphic solutions establishes an equivalence betweenthe bounded derived categories of complexes of D -modules with regular holo-nomic cohomology and the bounded derived category of complexes of sheaveswith constructible cohomology. In his proof ([13, 14]), M. Kashiwara gave anexplicit inverse to the functor of holomorphic solutions, the functor T H om .Kashiwara defined it on the category of R -constructible sheaves. The objectsof such a category satisfy the above conditions of locally constancy and finite-ness on some subanalytic stratification. For various reasons, the category of R -constructible sheaves is more handy than the category of C -constructiblesheaves. On the other hand the D -modules obtained by applying the functor T H om to R -constructible sheaves which are not C -constructible are still notwell understood.Later ([17, 18]), M. Kashiwara and P. Schapira deepened the study of thefunctor T H om (and its dual, the Whitney tensor) realizing it as the complexof sheaves on the subanalytic site X sa relative to the analytic manifold X oftempered holomorphic functions, denoted O tX sa . Furthermore, in [19], theysuggested the use of tempered holomorphic solutions of holonomic D -modulesfor approaching the general case of the Riemann–Hilbert correspondence.Since then, this latter idea produced significant contributions to the study ofirregular D -modules. Let us briefly recall the classical results in this subject.The local, 1-dimensional version of the irregular Riemann–Hilbert corre-spondence has been solved through the work of many mathematicians as P.Deligne, M. Hukuhara, A. Levelt, B. Malgrange, J.-P. Ramis, Y. Sibuya, H.Turrittin (see [7], [23] and [33]). Such a classical result consists in two steps:the analysis of the formal decompositions of meromorphic connections andtheir asymptotic lifts. The higher dimensional case is more complicated. Theformal decomposition was conjectured by C. Sabbah in [32] and later provedby T. Mochizuki ([24, 25]) in the algebraic case and by K. Kedlaya ([20],[21]) in the analytic one. Recently ([6, 5]), A. D’Agnolo and M. Kashiwaradefined the functor of enhanced tempered solutions using tempered holo-morphic functions. They proved that it is fully faithful on the category ofholonomic D X -modules and they proved a reconstruction theorem for suchcategory. Such a result is a generalization of the classical ones as it is ofglobal nature. Anyway, for the moment being, the statement of a Riemann–Hilbert correspondence for holonomic D -modules is not clear. Indeed, despiteof the important results characterizing the enhanced tempered solutions ofholonomic D -modules given in [5], a topological description (as for perversesheaves or C -constructible sheaves for the regular case) of the image categoryof the enhanced tempered solution functor has not been achieved.In [19], the authors give a definition of R -constructibility for complexes ofsheaves on the subanalytic sites and they conjectured the R -constructibilityfor tempered solutions of holonomic D -modules. This should be a first at-tempt to describe the image category of a functor of tempered solutions.The conjecture was proved for the case of complex curves ([26]) and in aweaker version for complex manifold ([27]). In the present article we provethe conjecture in its full generality.Let us conclude by recalling that Kashiwara–Schapira’s conjecture is strictlyrelated to Kashiwara’s functor T H om and the D -modules obtained by ap-plying it to R -constructible sheaves. The conjecture states that the the com-plex of solutions of these latter modules (which are not even coherent) withvalues in any holonomic D -module is a complex of sheaves on X with R -constructible cohomology.The present article is organized as follows. In the first section we re-view classical results on the subanalytic geometry, the sheaves on subanalyticsites, the D -modules, the tempered De Rham complexes and the elementaryasymptotic decompositions of flat meromorphic connections. In the secondsection we prove our main result in two steps, first we prove Kashiwara–Schapira’s conjecture for meromorphic connections, then for D -modules sup-ported on their singular locus. Acknowledgments : I wish to express my gratitude to M. Kashiwara, T.Mochizuki, C. Sabbah and P. Schapira for many essential discussions. Mostof the results exposed in this paper were obtained during my stay at RIMS,Kyoto University, I acknowledge the kind hospitality I found there.
Funding : the research leading to these results has received funding from the[European Union] Seventh Framework Programme [FP7/2007-2013] undergrant agreement n. [PIOF-GA-2010-273992].
For the theory of subanalytic sets we refer to [1, 11, 10]. For the convenienceof the reader we recall here some definitions and the Rectilinearization The-orem in a case which we will need later.Let M be a real analytic manifold. Definition 1.1.1. (i) A subset A of M is said subanalytic at p ∈ M ifthere exist a neighborhood W of p , a finite set J , real analytic manifolds N r,j and proper real analytic maps f r,j : N r,j → W ( r = 1 , , j ∈ J )such that A ∩ W = [ j ∈ J ( f ,j ( N ,j ) \ f ,j ( N ,j )) . A subset is called subanalytic if it is subanalytic at any point of M .(ii) A subset B of R n is called a quadrant if there exists a disjoint partition { , . . . , n } = J ⊔ J + ⊔ J − such that B = (cid:26) ( x , . . . , x n ) ∈ R n ; x j = 0 j ∈ J x k > k ∈ J + x l < l ∈ J − (cid:27) . Theorem 1.1.2 ([10], Chapters 4 and 7) . Let U be an open subanalyticsubset of a real analytic manifold M of dimension n and let ϕ : M → R bean analytic map which does not vanish on U . For any x ∈ M there exist areal analytic map π : R n → X and a compact set K ⊂ R n such that(i) π ( K ) is a neighborhood of x ∈ M ,(ii) π − ( U ) is a union of quadrants in R n ,(iii) π induces an open embedding of π − ( U ) into X ,(iv) ϕ ◦ π ( y , . . . , y n ) = a · y r · . . . · y r n n , the r j ’s being non-negative integersand a ∈ R \ { } . For the theory of sheaves on topological spaces and for the results onderived categories we will use, we refer to [16]. For the theory of sheaves onthe subanalytic site that we are going to recall now, we refer to [18] and [29].Let X be a real analytic manifold countable at infinity. The subanalyticsite X sa associated to X is defined as follows. An open subset U of X isan open set for X sa if it is relatively compact and subanalytic. The familyof open sets of X sa is denoted Op c ( X sa ). For U ∈ Op c ( X sa ), a subset S of the family of open subsets of U is said an open covering of U in X sa if S ⊂ Op c ( X sa ) and, for any compact K of X , there exists a finite subset S ⊂ S such that K ∩ ( ∪ V ∈ S V ) = K ∩ U .For Y = X or X sa , k a commutative ring, one denotes by k Y the constantsheaf. For a sheaf of rings R Y , one denotes by Mod( R Y ) the category ofsheaves of R Y -modules on Y and by D b ( R Y ) the bounded derived categoryof Mod( R Y ).With the aim of defining the category Mod( k X sa ), the adjective “relativelycompact” can be omitted in the definition of X sa . Indeed, in [18, Remark6.3.6], it is proved that Mod( k X sa ) is equivalent to the category of sheaves .1 Subanalytic sites X and whosecoverings are the same as X sa .One denotes by ̺ : X −→ X sa , the natural morphism of sites given by the inclusion of Op c ( X sa ) into thecategory of open subsets of X . We refer to [18] for the definitions of thefunctors ̺ ∗ : Mod( k X ) −→ Mod( k X sa ) and ̺ − : Mod( k X sa ) −→ Mod( k X )and for Proposition 1.1.3 below. Proposition 1.1.3. (i) The functor ̺ − is left adjoint to ̺ ∗ .(ii) The functor ̺ − has a left adjoint denoted by ̺ ! : Mod( k X ) → Mod( k X sa ) .(iii) The functors ̺ − and ̺ ! are exact, ̺ ∗ is exact on R -constructible sheaves.(iv) The functors ̺ ∗ and ̺ ! are fully faithful. The functor ̺ ! is described as follows. If U ∈ Op c ( X sa ) and F ∈ Mod( k X ),then ̺ ! ( F ) is the sheaf on X sa associated to the presheaf U F (cid:0) U (cid:1) .Let us conclude this subsection by recalling the definition of R -sa-constructibilityfor sheaves on X sa . It is due to M. Kashiwara and P. Schapira ([19]).Denote by D b R − c ( C X ) the full triangulated subcategory of D b ( C X ) consist-ing of complexes whose cohomology modules are R -constructible sheaves. Inwhat follows, for F ∈ D b ( C X sa ) and G ∈ D b R − c ( C X ), we set for short R H om C X ( G, F ) := ̺ − R H om C Xsa ( R̺ ∗ G, F ) ∈ D b ( C X ) . Definition 1.1.4.
Let F ∈ D b ( C X sa ) . We say that F is R subanalyticconstructible ( R -sa-constructible for short) if for any G ∈ D b R − c ( C X ) , (1.1) R H om C X ( G, F ) ∈ D b R − c ( C X ) . Let us remark that the condition (1.1), defining the property of R -sa-constructibility, implies that such property is local with respect to the topol-ogy of X sa and that it can be tested locally on X (i.e. with respect to G ). Remark 1.1.5.
The definition of R -sa-constructibility on X sa given above isquite abstract. A more geometrical description of such property, recalling theclassical one for sheaves on analytic manifolds, would be of great interest.With this aim in mind, it is worth recalling that, by means of classicaltopos theory, one can prove the existence of a topological space g X sa such that Mod( C g X sa ) is equivalent to Mod( C X sa ) . The semi-algebraic case is deeplystudied in [4]. Anyway, the different descriptions of g X sa do not allow astraightforward generalization of the classical constructibility property. D -modules The results on D -modules we are going to use in this paper are well exposed inthe literature, see for example [15] and [3]. Nonetheless, for the convenienceof the reader, we prefer to recall them here.Let X be a complex analytic manifold. One denotes by O X the sheaf ofrings of holomorphic functions on X and by D X the sheaf of rings of linearpartial differential operators with coefficients in O X .Given two left D X -modules M , M , one denotes by M D ⊗M the internaltensor product and by · D ⊗ · its extension to the derived category of D X -modules. Let us start by recalling the following Proposition 1.2.1.
Let
N ∈ D b ( D opX ) , M , M ∈ D b ( D X ) . Then N L ⊗ D X ( M D ⊗ M ) ≃ ( N L ⊗ O X M ) L ⊗ D X M . Now, let T ∗ X denote the cotangent bundle on X . We denote by Mod coh ( D X )the full subcategory of Mod( D X ) whose objects are coherent over D X . For M ∈
Mod coh ( D X ) we denote by char M the characteristic variety of M .Recall that char M ⊂ T ∗ X and that M is said holonomic if char M isLagrangian. One denotes by Mod h ( D X ) the full subcategory of Mod( D X )consisting of holonomic modules.One denotes by D bcoh ( D X ) (resp. D bh ( D X )) the full subcategory of D b ( D X )consisting of bounded complexes whose cohomology modules are coher-ent (resp. holonomic) D X -modules. For M ∈ D bcoh ( D X ), set char M := ∪ j ∈ Z char H j ( M ).Let π X : T ∗ X → X be the canonical projection, T ∗ X X the zero section of T ∗ X and ˙ T ∗ X := T ∗ X \ T ∗ X X .For M ∈ D bcoh ( D X ), the singular locus of M is defined as S ( M ) := π X (cid:16) char M ∩ ˙ T ∗ X (cid:17) . It is well known that, if
M ∈ D bh ( D X ), then S ( M ) = X is a closed analyticsubset of X .Now, we are going to recall the definition of a regular holonomic D -module.There are several equivalent definitions (see [15, Definition 5.2 and Proposi-tion 5.5]), we chose the following one because, in our opinion, it is the mostdirect and easy to state. Definition 1.2.2.
An object
M ∈ D bh ( D X ) is said regular holonomic if, forany x ∈ X , RHom D X ( M , O X,x ) ∼ −→ RHom D X ( M , b O X,x ) , .2 D -modules where b O X,x is the D X,x -module of formal power series at x . We denote by D brh ( D X ) the full subcategory of D bh ( D X ) consisting of M ∈ D bh ( D X ) all ofwhose cohomology modules are regular. Now, let Z be a closed analytic subset of X . Let I Z be the coherent idealconsisting of the holomorphic functions vanishing on Z , for M ∈
Mod( D X ),one sets Γ [ Z ] M := lim −→ k H om O X ( O X / I kZ , M ) , Γ [ X \ Z ] M := lim −→ k H om O X ( I kZ , M ) . If S ⊂ X can be written as S = Z \ Z , for Z and Z closed analytic sets,then it can be proved that the following object is well definedΓ [ S ] M := Γ [ Z ] Γ [ X \ Z ] M and that Γ [ S ] is a left exact functor. One denotes by R Γ [ S ] the right derivedfunctor of Γ [ S ] . Theorem 1.2.3.
Let Z be a closed analytic subset of X , S , S differencesof closed analytic subsets of X , M ∈ D b ( D X ) .(i) The following is a distinguished triangle in D b ( D X ) R Γ [ Z ] M −→ M −→ R Γ [ X \ Z ] M +1 −→ . (ii) We have R Γ [ Z ] M ≃ R Γ [ Z ] O D ⊗ M and R Γ [ X \ Z ] M ≃ R Γ [ X \ Z ] O D ⊗ M .(iii) We have R Γ [ S ] R Γ [ S ] M ≃ R Γ [ S ∩ S ] M . Given f ∈ O X , let Z := f − (0). One denotes by O X [ ∗ Z ] the sheaf ofmeromorphic functions with poles on Z . Let us remark that O X [ ∗ Z ] isflat over O X . Given M ∈
Mod( D X ), one can prove that R Γ [ X \ Z ] M ≃M D ⊗ O X [ ∗ Z ].Given two complex analytic manifolds X , Y and a holomorphic map f : Y → X , one denotes by D f ∗ : Mod( D X ) → Mod( D Y ) the inverse imagefunctor and by D f ∗ : D b ( D X ) → D b ( D Y ) the derived functor. Moreover,one has D f ∗ : D bh ( D X ) → D bh ( D Y ). Recall that f is said smooth if thecorresponding maps of tangent spaces T y Y → T f ( y ) X are surjective for any y ∈ Y . If f is a smooth map, then D f ∗ is an exact functor.We conclude describing the behavior of R Γ [ X \ Z ] with respect to D f ∗ .Proposition 1.2.4 below can be directly obtained using, for example, Propo-sition 2.5.27 and Theorem 2.3.17 of [3]. Proposition 1.2.4.
Let f : Y → X be a holomorphic map, Z an analyticsubset of X , M ∈ D bh ( D X ) . Then, D f ∗ R Γ [ X \ Z ] M ≃ R Γ [ Y \ f − ( Z )] D f ∗ M . We start this subsection by recalling some results of [18] on temperedholomorphic functions and tempered De Rham complexes of holonomic D -modules. Then we prove some general results on the R -sa-constructibility ofthe tempered De Rham complex with respect to the inverse image functors.Given a complex analytic manifold X of dimension d X , one denotes by X R the real analytic manifold underlying X . Furthermore, one denotes by X the complex conjugate manifold, in particular O X is the sheaf of anti-holomorphic functions on X . Denote by D b X R the sheaf of distributions on X R and, for a closed subset Z of X , by Γ Z ( D b X R ) the subsheaf of sectionssupported by Z . One denotes by D b tX sa the presheaf of tempered distributions on X sa defined byOp c ( X sa ) op ∋ U b tX sa ( U ) := Γ( X ; D b X R ) (cid:14) Γ X \ U ( X ; D b X R ) . In [18] it is proved that D b tX sa is a sheaf on X sa . This sheaf is well definedin the category Mod( ̺ ! D X ). Moreover, for any U ∈ Op c ( X sa ), D b tX sa isΓ( U, · )-acyclic.The sheaf D b tX sa is strictly related to the Kashiwara’s T H om ( · , D b X ) func-tor introduced in [13] and deeply studied [14]. For the definition see [14,Definition 3.13]. Let us simply recall [18, Proposition 7.2.6 (i) ](1.2) R H om( F, D b tX sa ) ≃ T H om ( F, D b X ) , for F ∈ D b R − c ( C X ).One defines the complex of sheaves O tX sa ∈ D b (cid:0) ̺ ! D X (cid:1) of tempered holo-morphic functions as(1.3) O tX sa := R H om ̺ ! D X (cid:0) ̺ ! O X , D b tX sa (cid:1) . It is worth to mention that, if dim X = 1, then O tX sa is concentrated indegree 0 and, for U ∈ Op c ( X sa ), we have O tX sa ( U ) ≃ { u ∈ O X ( U ); ∃ C, N > , ∀ x ∈ U, | u ( x ) | ≤ C dist( x, ∂U ) − N } . We also have [18, Proposition 7.3.2](1.4) R H om( F, O tX sa ) ≃ T H om ( F, O X ) , for F ∈ D b R − c ( C X ).Let Ω jX be the sheaf of differential forms of degree j . For sake of simplicity,let us write Ω X instead of Ω d X X .Set Ω tX := ̺ ! Ω X ⊗ ̺ ! O X O tX sa . .3 Tempered De Rham complexes Theorem 1.3.1 ([18] Theorem 7.4.12, Theorem 7.4.1) . (i) Let L ∈ D brh ( D X ) and set L := R H om D X ( L , O X ) . There exists anatural isomorphism in D b ( C X sa )Ω tX L ⊗ ̺ ! O X ̺ ! L ≃ R H om C Xsa ( L, Ω tX ) . (ii) Let X, Y be complex manifolds of dimension, respectively, d X and d Y ; f : Y → X a holomorphic map and let N ∈ D b ( D X ) . There is anatural isomorphism in D b ( C Y sa )Ω tY L ⊗ ̺ ! D Y ̺ ! ( D f ∗ N )[ d Y ] ∼ −→ f ! (Ω tX L ⊗ ̺ ! D X ̺ ! N )[ d X ] . For
M ∈ D b ( D X ), we set for shortDR tX M := Ω tX L ⊗ ̺ ! D X ̺ ! M [ − d X ] ∈ D b ( C X sa ) , S ol t ( M ) := R H om ̺ ! D X ( ̺ ! M , O tX sa ) ∈ D b ( C X sa ) . For
M ∈
Mod h ( D X ), set D X M := E xt d X D X ( M , D X ⊗ O X Ω ⊗− X ) . Proposition 1.3.2. (i) The functor D X : Mod h ( D X ) op → Mod h ( D X ) isan equivalence of categories.(ii) Let M ∈ D bh ( D X ) . Then, (1.5) S ol t ( M ) ≃ DR tX ( D X M ) . We are now going to recall a conjecture of M. Kashiwara and P. Schapiraon the R subanalytic constructibility of tempered holomorphic solutions ofholonomic D -modules and the result we obtained on curves. Conjecture 1.3.3 ([19]) . Let
M ∈ D bh ( D X ) . Then S ol t ( M ) ∈ D b ( C X sa ) is R -sa-constructible. In [26], we proved that Conjecture 1.3.3 is true on analytic curves.
Theorem 1.3.4.
Let X be a complex curve and M ∈ D bh ( D X ) . Then, S ol t ( M ) is R -sa-constructible. Later we will use a version of the conjecture using the functor
T H om instead of the complex of sheaves O t . Let us show how to pass from one tothe other with the following easy Lemma 1.3.5.
For G ∈ D b R − c ( C X ) and M ∈ D bh ( D X ) we have (1.6) R H om C Xsa ( G, S ol t ( M )) ≃ R H om D X ( M , T H om C Xsa ( G, O X )) . Proof.
We have the following sequence of isomorphisms ̺ − R H om C Xsa ( ̺ ∗ G, R H om ̺ ! D X ( ̺ ! M , O tX sa )) ≃≃ ̺ − R H om ̺ ! D X ( ̺ ∗ G ⊗ C Xsa ̺ ! M , O tX sa ) ≃ ̺ − R H om ̺ ! D X ( ̺ ! M , R H om C Xsa ( ̺ ∗ G, O tX sa )) ≃ R H om D X ( M , ̺ − R H om C Xsa ( ̺ ∗ G, O tX sa )) ≃ R H om D X ( M , T H om C Xsa ( G, O X )) , where we used [29, Proposition 1.1.16] in the third isomorphism.As it will be useful later, let us give an explicit form to R H om C X ( C U , DR tX M ), for U ∈ Op c ( X sa ), M ∈
Mod h ( D X ).Recall that one denotes by X R the real analytic manifold underlying X , by C ωX R the sheaf of real analytic functions on X R , by X the complex conjugatemanifold of X , by O X (resp. D X , Ω X ) the sheaf of holomorphic functions(resp. linear differential operators with coefficients in O X , maximal degreedifferential forms) on X . Furthermore, we denote by Ω • X (resp. Ω • , • X R ) thecomplex (resp. double complex) of sheaves of differential forms on X (resp. X R ) with coefficients in {O X (resp. C ωX R ).Now, since R H om C X ( C U , DR tX M ) ≃ R H om D X R ( D M ⊗ O X C ω , T H om ( C U , D b X )) ≃ T H om ( C U , D b X ) ⊗ C ω Ω • , • X R ⊗ O X D M , we have that R H om C X ( C U , DR tX M ) is isomorphic to the total complex rel-ative to the double complex(1.7) 0 (cid:15) (cid:15) (cid:15) (cid:15) / / T H om ( C U , D b X ) ⊗ C ωX R Ω , X R ⊗ O X D M ∇ / / ∂ (cid:15) (cid:15) T H om ( C U , D b X ) ⊗ C ωX R Ω , X R ⊗ O X D M ∇ / / ∂ (cid:15) (cid:15) . . . / / T H om ( C U , D b X ) ⊗ C ωX R Ω , X R ⊗ O X D M ∇ / / ∂ (cid:15) (cid:15) T H om ( C U , D b X ) ⊗ C ωX R Ω , X R ⊗ O X D M ∇ / / ∂ (cid:15) (cid:15) . . .. . . . . . where the ∇ above, in a local coordinate system z : V → C n , is defined by ∇ ( f ⊗ ω ⊗ m ) = n X j =1 ∂ z j f ⊗ dz j ∧ ω ⊗ m + f ⊗ dω ⊗ m + f ⊗ n X j =1 dz j ∧ ω ⊗ ∂ z j m .3 Tempered De Rham complexes f ∈ D b tX sa , ω ∈ Ω k,hX R and m ∈ D M .Now, we prove some general results on the behaviour of the R -sa-constructibility of a tempered De Rham complex with respect to the tensorproduct and the inverse image functor. We obtained similar results in [27],we adapted the proofs to the present case. Lemma 1.3.6.
Let
M ∈ D bh ( D X ) and R ∈ D brh ( D X ) . Set L := R H om D X ( R , O X ) ∈ D b R − c ( C X ) . For any G ∈ D b R − c ( C X ) , one has that (1.8) R H om C X ( G, DR tX ( M D ⊗ R )) ≃ R H om C X ( G ⊗ C X L, DR tX M ) In particular, if DR tX M is R -sa-constructible, then DR tX ( M D ⊗ R ) is R -sa-constructible.Proof. Let G ∈ D b R − c ( C X ), the following sequence of isomorphisms proves(1.8) R H om C X ( G, Ω tX L ⊗ ̺ ! D X ̺ ! ( M D ⊗ R )) ≃ R H om C X ( G, (Ω tX L ⊗ ̺ ! O X ̺ ! R ) L ⊗ ̺ ! D X ̺ ! M ) ≃ R H om C X ( G, R H om C Xsa ( L, Ω tX ) L ⊗ ̺ ! D X ̺ ! M ) ≃ R H om C X ( G, R H om C Xsa ( L, Ω tX L ⊗ ̺ ! D X ̺ ! M )) ≃ R H om C X ( G ⊗ C X L, Ω tX L ⊗ ̺ ! D X ̺ ! M ) . In the previous series of isomorphisms we used Proposition 1.2.1 in thefirst isomorphism and Theorem 1.3.1(i) in the second isomorphism.Let us recall two definitions.
Definition 1.3.7. (i) Given a closed analytic set Z ⊂ X , a morphism f : Y → X of analytic manifolds is said to be a modification withrespect to Z , or simply a modification, if it is proper and if f | Y \ f − ( Z ) is an isomorphism on X \ Z .(ii) Suppose that X ≃ C n and Z is defined by the equation x · . . . · x k = 0 .By a ramification map fixing Z we mean a map ̺ l : C n −→ C n ( t . . . , t n ) ( t l . . . , t lk , t k +1 . . . , t n ) , for some l ∈ Z > . Proposition 1.3.8. (i) Given Z = X a closed analytic hypersurface of X , f : Y → X a modification with respect to Z , M ∈ D b ( D X ) suchthat R Γ [ X \ Z ] M ≃ M and G ∈ D b R − c ( C X ) , we have (1.9) R H om C X ( G, DR tX M ) ≃≃ Rf ∗ R H om C Y ( f − ( G ⊗ C X \ Z ) , DR tY D f ∗ M ) . In particular, if DR tY D f ∗ M is R -sa-constructible, then DR tX M is R -sa-constructible.(ii) Suppose we are in the situation of Definition 1.3.7 (ii). Let f : Y → X be a ramification fixing Z , M ∈ D b ( D X ) such that R Γ [ X \ Z ] M ≃ M , U ∈ Op c ( X sa ) . There exists a local system L on X \ Z such that thefollowing isomorphism holds (1.10) Rf ∗ R H om C Y ( f − ( C U \ Z ) , DR tY D f ∗ M ) ≃≃ R H om C X ( C U ⊕ L, DR tX M ) . In particular, if DR tY D f ∗ M is R -sa-constructible, then DR tX M is R -sa-constructible.Proof. (i) Let f : Y → X be a modification. The following sequence ofisomorphisms proves the statement R H om C X ( G, Ω tX L ⊗ ̺ ! D X ̺ ! R Γ [ X \ Z ] M ) ≃≃ R H om C X ( G, Ω tX L ⊗ ̺ ! D X ̺ ! ( R Γ [ X \ Z ] O X D ⊗M )) ≃ R H om C X ( G ⊗ C X \ Z , Ω tX L ⊗ ̺ ! D X ̺ ! M ) ≃ R H om C X ( Rf ! f − ( G ⊗ C X \ Z ) , Ω tX L ⊗ ̺ ! D X ̺ ! M ) ≃ Rf ∗ R H om C Y ( f − ( G ⊗ C X \ Z ) , f ! (Ω tX L ⊗ ̺ ! D X ̺ ! M )) ≃ Rf ∗ R H om C Y ( f − ( G ⊗ C X \ Z ) , Ω tY L ⊗ ̺ ! D Y ̺ ! ( D f ∗ M )) . We have used Lemma 1.3.6 with R := R Γ [ X \ Z ] O X in the second isomor-phism, the fact that f | Y \ f − ( Z ) is an isomorphism in the third isomorphismand Theorem 1.3.1 (ii) in the last isomorphism. (ii) There exists a local system L on U \ Z such that Rf ! f − C U \ Z ≃ C U \ Z ⊕ L . We have .4 Asymptotic decomposition of meromorphic connections Rf ∗ R H om C Y ( f − ( C U \ Z ) , Ω tY L ⊗ ̺ ! D Y ̺ ! ( D f ∗ M )) ≃≃ Rf ∗ R H om C Y ( f − ( C U \ Z ) , f ! (Ω tX L ⊗ ̺ ! D X ̺ ! M )) ≃ R H om C X ( Rf ! f − ( C U \ Z ) , Ω tX L ⊗ ̺ ! D X ̺ ! M ) ≃ R H om C X ( C U \ Z ⊕ L, Ω tX L ⊗ ̺ ! D X ̺ ! M ) ≃ R H om C X ( C U ⊕ L, Ω tX L ⊗ ̺ ! D X ̺ ! ( R Γ [ X \ Z ] O X D ⊗M )) ≃ R H om C X ( C U ⊕ L, Ω tX L ⊗ ̺ ! D X ̺ ! R Γ [ X \ Z ] M ) ≃ R H om C X ( C U ⊕ L, Ω tX L ⊗ ̺ ! D X ̺ ! M ) , where we used Theorem 1.3.1 (ii) in the first isomorphism, Lemma 1.3.6with R := R Γ [ X \ Z ] O X in the fourth isomorphism and Theorem 1.2.3 (ii) inthe fifth isomorphism.In Subsection 2.1 we will need also some results on D M -modules for M areal analytic manifold, i.e. modules over the sheaf of rings of linear differentialoperators of finite order with real analytic coefficients on M . We refer to [3,Section VII.1]. Let us conclude with some notation about D M -modules andtheir De Rham complexes. We denote with Ω ω the sheaf on M of differentialforms of maximal degree with real analytic coefficients. For M ∈
Mod( D M ),we set DR tM M := ̺ ! M L ⊗ ̺ ! D M (Ω ω ⊗ C ω D b tM s a ) . Even for such an object there are formulae similar to (1.6), (1.9), (1.10) andthose in the statement of Theorem 1.3.1 (see [30, 28]).
In this subsection we are going to recall some fundamental results on theasymptotic decompositions of flat meromorphic connections. The first re-sults on this subject were obtained by H. Majima ([22]) and C. Sabbah([31]). In particular, Sabbah proved that any good formal decomposition of aflat meromorphic connection on a complex surface admits an asymptotic lifton small multisectors. Let us recall that Sabbah conjectured that any flatmeromorphic connection admits a good formal decomposition after a finite4 number of complex pointwise blow-up and ramification maps ([32]). Such aconjecture was proved in the algebraic case by T. Mochizuki ([24, 25]) and inthe analytic one by K. Kedlaya ([20, 21]). As in this paper we are not con-cerned with the formal decomposition of flat meromorphic connections andsince the goodness property is not needed within the scope of our results, weare not going to give details on them. In this way, we will avoid to go intounessential technicalities for the rest of the paper. For this subsection werefer to [32, 25, 8].Let X be an analytic manifold of dimension n and Z a divisor of X . Letus start by recalling some results about integrable connections on X withmeromorphic poles on Z . As in the rest of the paper we will just need thecase where Z is a normal crossing hypersurface, from now on, we will supposesuch hypothesis.Let Ω jX be the sheaf of j -forms on X . Let M be a finitely generated O X [ ∗ Z ]-module endowed with a C X -linear morphism ∇ : M → Ω X ⊗ O X M satisfying the Leibniz rule, that is to say, for any h ∈ O X [ ∗ Z ], m ∈ M , ∇ ( hm ) = dh ⊗ m + h ∇ m . The morphism ∇ induces C X -linear morphisms ∇ ( j ) : Ω jX ⊗ O X M → Ω j +1 X ⊗ O X M . Definition 1.4.1. A flat meromorphic connection on X with poles along Z is a locally free O X [ ∗ Z ] -module of finite type M endowed with a C X -linearmorphism ∇ as above such that ∇ (1) ◦ ∇ = 0 . In general, in the literature, a meromorphic connection is just a coherent O [ ∗ Z ]-module. As stated in Proposition 1.2.1 of [32], locally, the condition ofbeing locally free is not restrictive. Since in this paper we deal with the localstudy of meromorphic connections and holonomic D -modules, we adoptedDefinition 1.4.1. For sake of shortness, in the rest of the paper, we will dropthe adjective “ flat ”. If there is no risk of confusion, given a meromorphicconnection ( M , ∇ ), we will simply denote it by M .Let M , M be two locally free O X [ ∗ Z ]-modules, a morphism ϕ : M →M induces a morphism ϕ ′ : Ω X ⊗ O X M → Ω X ⊗ O X M . A morphismof meromorphic connections ( M , ∇ ) → ( M , ∇ ) is given by a morphism ϕ : M → M of locally free O X [ ∗ Z ]-modules such that ϕ ′ ◦ ∇ = ∇ ◦ ϕ .We denote by M ( X, Z ) the category of meromorphic connections with polesalong Z .Let us recall some facts on meromorphic connections that will be useful inthis paper, we refer to [3] or Sections I.1.2 and I.1.3 of [32].It is well known that the category M ( X, Z ) is equivalent to the image inMod h ( D X ) of the functor · D ⊗ O [ ∗ Z ] on the full subcategory of Mod h ( D X )consisting of objects with singular locus contained in Z . In particular, if M ∈
Mod h ( D X ) is such that S ( M ) is contained in Z and M ≃ R Γ [ X \ Z ] M ≃M D ⊗ O [ ∗ Z ], then M is a locally free O X [ ∗ Z ]-module and the morphism ∇ : .4 Asymptotic decomposition of meromorphic connections M −→ Ω X ⊗ O X M , defined in a local coordinate system z : U → C n by ∇ m := n P j =1 dz j ⊗ ∂ z j m , gives rise to a meromorphic connection. A meromorphicconnection is said regular if it is regular as a D X -module. Furthermore thetensor product in M ( X, Z ) is well defined and it coincides with the tensorproduct of D X -modules. With an abuse of language, given a holonomic D X -module M with singular locus contained in Z , we will call M a meromorphicconnection if M ≃ R Γ [ X \ Z ] M ≃ M D ⊗ O X [ ∗ Z ].Let us recall that, if ( M , ∇ ) ∈ M ( X, Z ) and M is an O X [ ∗ Z ]-module ofrank r , then, in a given basis of local sections of M , we can write ∇ as d − A where A is a r × r matrix with entries in Ω X ⊗ O X O X [ ∗ Z ]. Now, let X ′ be acomplex manifold and f : X ′ → X a holomorphic map. Let us suppose that Z ′ := f − ( Z ) is a normal crossings hypersurface of X ′ and that f is smoothon X ′ \ Z ′ . Considering M as a D X -module, it satisfies R Γ [ X \ Z ] M ≃ M .Moreover, since, for j ≥
1, supp H j D f ∗ M ⊂ Z ′ , one checks that D f ∗ M ≃ D f ∗ M ≃ R Γ [ X ′ \ Z ′ ] D f ∗ M . Hence D f ∗ M can be considered as an object of M ( X ′ , Z ′ ). As an O X ′ [ ∗ Z ′ ]-module, it is isomorphic to f − M and the matrixof the connection in a local base is f ∗ A .Let us now introduce the elementary asymptotic decompositions.Let us denote by e X the real oriented blow-up of the irreducible componentsof Z and by π : e X → X the composition of all these. Let us suppose that Z is locally defined by x · . . . · x k = 0. Then, locally e X ≃ ( S × R ≥ ) k × C n − k .In what follows, S will be identified with the unit circle in C , so S = { e iϑ ∈ C ; ϑ ∈ R } .A multisector is an open subset S ⊂ e X such that there exist a j , b j ∈ R , a j < b j , r j ∈ R > ( j = 1 , . . . , k ), W an open neighborhood of 0 ∈ C n − k suchthat(1.11) S = (cid:16) k Y j =1 I j × [0 , r j [ (cid:17) × W ⊂ ( S × R ≥ ) k × C n − k , where I j = { e iϑ ∈ C ; ϑ ∈ ] a j , b j [ } ⊂ S . Given ( r, τ ) ∈ R > , we say that amultisector S is small with respect to ( r, τ ) if S can be written as in (1.11)and, for any j = 1 , . . . , k , b j − a j < τ , r j < r and for any x ∈ W , | x | < r .Let x , . . . , x n denote the antiholomorphic coordinates on X . Then, for j = k + 1 , . . . , n (resp. j = 1 , . . . , k ) ∂ x j (resp. x j ∂ x j ) acts on C ∞ e X (see [31]2.12 or [32] 1.1.4). The sheaf of algebras on e X of holomorphic functions withasymptotic development on Z , denoted A e X , is defined as A e X := k \ j =1 ker (cid:16) x j ∂ x j : C ∞ e X → C ∞ e X (cid:17) ∩ n \ j = k +1 ker (cid:16) ∂ x j : C ∞ e X → C ∞ e X (cid:17) . The sections of A e X are holomorphic functions on e X which admit an asymp-totic development as explained in Proposition B.2.1 of [32]. Moreover, A e X is a π − O X -module. Definition 1.4.2. (i) Let ϕ be a local section of O X [ ∗ Z ] / O X , we denoteby L ϕ the meromorphic connection of rank whose matrix in a basisis dϕ .(ii) An elementary local model M is a meromorphic connection isomorphicto a direct sum ⊕ α ∈ A L ϕ α ⊗ R α , where A is a finite set, ( ϕ α ) α ∈ A is a family of local sections of O X [ ∗ Z ] / O X and ( R α ) α ∈ A is a family of regular meromorphic connec-tions.(iii) We say that ( M , ∇ ) ∈ M ( X, Z ) admits an elementary A -decomposition if there exist an elementary local model ( M el , ∇ el ) and ( r, τ ) ∈ R > such that for any multisector S ⊂ e X small with respectto ( r, τ ) , there exists Y S ∈ G l (rk M , A e X ( S )) such that the followingdiagram commutes (1.12) π − M ⊗ π − O X A e X ( S ) ∇ / / ∼ Y S · (cid:15) (cid:15) π − ( M ⊗ O X Ω X ) ⊗ π − O X A e X ( S ) ∼ Y S · (cid:15) (cid:15) π − M el ⊗ π − O X A e X ( S ) ∇ el / / π − ( M el ⊗ O X Ω X ) ⊗ π − O X A e X ( S ) . Remark that the isomorphism in (1.12) depends on S and, in general, itdoes not give a global isomorphism.The following Theorem is obtained by combining fundamental results ofC. Sabbah ([31]), T. Mochizuki ([24, 25]) and K. Kedlaya ([20, 21]). Theorem 1.4.3.
Let ( M , ∇ ) ∈ M ( X, Z ) . For any x ∈ Z there exist aneighborhood W of x and a modification with respect to W ∩ Z above x , σ : Y → X such that σ − ( Z ) is a normal crossing divisor and there exists aramification map η : X ′ → Y fixing σ − ( Z ) such that D( η ◦ σ ) ∗ M| W admitsan elementary A -decomposition. R -sa-constructibility of the tempered DeRham complex of holonomic D -modules In this section we are going to prove the following .1 The case of R Γ [ X \ S ( M )] M Theorem 2.0.1.
Let X be a complex analytic manifold, M ∈ D bh ( D X ) .Then DR tX M ∈ D b ( C X sa ) is R -sa-constructible.Proof. By using the distinguished triangle R Γ [ S ( M )] M −→ M −→ R Γ [ X \ S ( M )] M +1 −→ , it turns out that it is sufficient to prove the R -sa-constructibility ofDR tX ( R Γ [ S ( M )] M ) and DR tX ( R Γ [ X \ S ( M )] M ). We will deal these two casesseparately in the next subsections. R Γ [ X \ S ( M )] M In the present Subsection, first we prove the conjecture for elementary mod-els, then for connections admitting an elementary A -decomposition and inthe end for meromorphic connections.Recall that, for Z a normal crossings hypersurface and ϕ ∈ O X [ ∗ Z ] / O X ,we denote by L ϕ the meromorphic connection of rank 1 whose matrix ina basis is dϕ . As a D X -module, L ϕ is isomorphic to R Γ [ X \ Z ] ( D X exp( ϕ )).Remark that, with a harmless abuse, we use the same notation for the realanalytic case. Lemma 2.1.1.
Consider X := C n with standard coordinates ( x , . . . , x n ) .Set Z := { ( x , . . . , x n ) ∈ X ; x · . . . · x n = 0 } . Given ϕ ∈ O ( ∗ Z ) O and R ∈ D brh ( D X ) , we have that DR tX ( L ϕ ⊗ R ) is R -sa-constructible.Proof. First, let us remark that, by Lemma 1.3.6 it is sufficient to prove thestatement with
R ≃ O X .Given U ∈ Op c ( C nsa ), by Theorem 1.1.2 we have that for any x ∈ R n there exist real analytic map g : R n → R n and a compact set K ⊂ R n suchthat(i) g ( K ) is a neighborhood of x ,(ii) g − ( U ) is a union of quadrants,(iii) g induces an open embedding of g − ( U ) into X .(iv) ψ ( y ) := ϕ ◦ g ( y ) = a · y − k · . . . · y − k n n , a ∈ R × , k , . . . , k n ∈ N .Since U is compact, it is sufficient to prove that R H om C X ( C U ∩ g ( K ) , DR tX L ϕ ) ∈ D b R − c ( C X ) . Set g k := g | K . We have the following sequence of isomorphisms(2.1) R H om C X ( C U ∩ g ( K ) , DR tX L ϕ ) ≃ ≃ R H om C X ( Rg K ! g − K C U ∩ g ( K ) , DR tX L ϕ ) ≃ Rg K ∗ R H om C K ( g − K C U ∩ g ( K ) , g ! K DR tX L ϕ ) ≃ Rg K ∗ R H om C K ( C g − ( U ) ∩ K , DR t R n D g K ∗ L ϕ ) ≃ Rg K ∗ R H om C K ( C g − ( U ) ∩ K , DR t R n L ψ ) . Set M := R n , V := g − ( U ) ∩ K and Z ′ := { ( y , . . . , y n ); y · . . . · y n = 0 } .Clearly one has R H om C M ( C V , DR tM L ψ ) M \ Z ′ ∈ D b R − c ( C M ) . Let us now consider, for y ∈ Z ′ R j H om C M ( C V , DR tM L ψ ) y . We are going to prove that it does not depend on y , this will conclude theproof.First, remark that R H om C M ( C V , DR tM L ψ ) ≃ R H om ̺ ! D M ( C V ⊗ C M \ Z ′ ⊗ D M L ψ , D b tM ) ≃ R H om ̺ ! D M ( L − ψ , R H om( C V , D b tM )) ≃ R H om D M ( L − ψ , T H om C V D b M ) . Now, R H om D M ( L − ψ , T H om ( C V , D b M )) y ≃≃ lim −→ y ∈ W R Hom ̺ ! D W ( j − W L − ψ , j − W T H om ( C V , D b M )) ≃ lim −→ y ∈ W R Hom ̺ ! D M ( L − ψ , Rj W ∗ j − W T H om ( C V , D b M )) ≃ lim −→ y ∈ W R Hom ̺ ! D M ( L − ψ , T H om ( C V ∩ W , D b M )) ≃ lim −→ y ∈ W R Hom C M ( C V ∩ W , DR tM L ψ ) , where in the third isomorphism, we used the fact that, for W ′ ⊂⊂ W , therestriction morphism T H om ( C V ∩ W , D b M ) −→ T H om ( C V ∩ W ′ , D b M )factorizes through Rj W ∗ j − W T H om ( C V , D b M ), for j W : W → M the inclusion.Consider the map f : M → R , f ( y , . . . , y n ) = a − y k · . . . · y k n n and set B := B ( y , ǫ ). We have the following series of isomorphisms .1 The case of R Γ [ X \ S ( M )] M R Hom C M ( C V ∩ B , DR tM L ψ ) ≃ R Hom C M ( C V ∩ B , DR tM D f ∗ L /t ) ≃ R Hom C M ( C V ∩ B , f ! DR t R L /t [1 − n ]) ≃ R Hom C R ( Rf ! C V ∩ B [2 n − , DR t R L /t ) ≃ R Hom C R ( C f ( V ∩ B ) , DR t R L /t ) ≃ R Hom C R ( L − /t , T H om ( C f ( V ∩ B ) , D b M )) . Using the well known solvability of the homogeneous and non homogeneousdifferential equations related to the operator t ddt − ǫ small enough the followingisomorphisms hold R j Hom D M ( L − ψ , T H om ( C V ∩ B ( y ,ǫ ) , D b M )) ≃≃ j >
00 if j = 0 and exp( − ψ ) / ∈ T H om ( C V ∩ B ( y ,ǫ ) , D b M ) C if j = 0 and exp( − ψ ) ∈ T H om ( C V ∩ B ( y ,ǫ ) , D b M ) . Let us now consider the case of meromorphic connections admitting anelementary A -decomposition. Lemma 2.1.2.
Let X be a complex analytic manifold and M ∈
Mod h ( D X ) be such that(i) S ( M ) is a normal crossing hypersurface,(ii) M ≃ R Γ [ X \ S ( M )] M ,(iii) as a meromorphic connection, M admits an elementary A -decomposition.Then DR tX M is R -sa-constructible.Proof. For sake of shortness, let us set Z := S ( M ).We want to prove that for any G ∈ D b R − c ( C X )(2.2) R H om C Xsa ( G, DR tX M ) ∈ D b R − c ( C X ) . First, let us remark that it is sufficient to prove (2.2) for G = C U for U ∈ Op c ( X sa ).We have the following sequence of isomorphisms0 R H om C X ( C U , Ω tX L ⊗ ̺ ! D X ̺ ! R Γ [ X \ Z ] M ) ≃ R H om C X ( C U , Ω tX L ⊗ ̺ ! D X ̺ ! ( O X [ ∗ Z ] D ⊗ M )) ≃ R H om C X ( C U ⊗ C X \ Z , Ω tX L ⊗ ̺ ! D X ̺ ! M ) . Where we used Lemma 1.3.6 with R := O X [ ∗ Z ] in the second isomorphism.Hence, it is sufficient to prove the (2.2) for G = C V , V ∈ Op c ( X sa ), V ⊂ X \ Z .Let us briefly recall the meaning of the hypothesis (iii) of the statementwe are proving. The problem being local, we can suppose that X ≃ C n and Z ≃ { ( x , . . . , x n ) ∈ C n ; x · . . . · x k = 0 } . Furthermore, as M ≃ R Γ [ X \ Z ] M and as V ⊂ X \ Z we can suppose that both M and D M are meromorphicconnections, in particular they are locally free O [ ∗ Z ]-modules of finite rank.Let Ω • X be the complex of differential forms on X .Recall that we defined e X as the real oriented blow-up of the irreduciblecomponents of Z and by π : e X → X the composition of all these. Then,locally e X ≃ ( S × R ≥ ) k × C n − k . Now, as M has an elementary A -decomposition, there exist an elementary model ( M el , ∇ el ) and ( r, τ ) ∈ R > such that for any multisector S ⊂ e X small with respect to ( r, τ ) there exists Y S ∈ G l (rk M , A e X ( S )) giving an isomorphism of complexes0 / / π − M ⊗ π − O X A e X ( S ) ∇ / / Y S ·∼ (cid:15) (cid:15) π − ( M ⊗ O X Ω X ) ⊗ π − O X A e X ( S ) ∇ / / Y S ·∼ (cid:15) (cid:15) . . . / / π − M el ⊗ π − O X A e X ( S ) ∇ el / / π − ( M el ⊗ O X Ω X ) ⊗ π − O X A e X ( S ) ∇ el / / . . . . Clearly, for any ( r, τ ) ∈ R > , there exists a finite family { S k } k ∈ K of mul-tisectors small with respect to ( r, τ ) such that, for any L ⊂ K , ∩ l ∈ L S l is amultisector small with respect to ( r, τ ) and ∪ k ∈ K S k is an open neighborhoodof π − ( Z ) ⊂ e X .Coming back to the proof of (2.2), as said above it is sufficient to proveit for G = C V , V ∈ Op c ( X sa ), V ⊂ X \ Z . Given such a V , consider V k := V ∩ π ( S k ) for { S k } k ∈ K a family of multisectors as above, then thereexists W ∈ Op c ( X sa ) such that W ∩ Z = ∅ and V = ∪ k ∈ K V k ∪ W . Now, sincefor any L ⊂ K , ∩ l ∈ L V l is contained in a multisector small with respect to( r, τ ), it follows that it is sufficient to prove (2.2) for G = C V , V ∈ Op c ( X sa ), V ⊂ X \ Z and π − ( V ) contained in a multisector S small with respect to( r, τ ).Recall that we denote by X R the real analytic manifold underlying X , by C ωX R the sheaf of real analytic functions on X R . Furthermore, we denote by .1 The case of R Γ [ X \ S ( M )] M • , • X R the double complex of sheaves of differential forms on X R with coeffi-cients in C ωX R .In Subsection 1.3, in (1.7), we proved that R H om C X ( C V , DR tX M ) is iso-morphic to the total complex relative to (1.7).Now, as T H om ( C π ( S ) \ Z , D b ) is a π ∗ A e X | S -module, it makes sens to consider Y S as an isomorphism T H om ( C V , D b ) ⊗ C ωX R Ω j,kX R ⊗ O X D M Y S · −→ ∼ T H om ( C V , D b ) ⊗ C ωX R Ω j,kX R ⊗ O X D M el . which satisfies the natural commuting conditions with respect to ∇ , ∇ el and ∂ . It follows that the total complex of (1.7) is isomorphic to the total complexrelative to the double complex obtained from (1.7) replacing M with M el and ∇ with ∇ el .The total complex of this last double complex is isomorphic to R H om C X ( C V , DR tX ( M el )).By Lemma 2.1.1, DR tX ( M el ) is R -sa-constructible. Hence we have thatDR tX M is R -sa-constructible too. Proposition 2.1.3.
Let X be a complex analytic manifold, M ∈ D bh ( D X ) .Then DR tX ( R Γ [ X \ S ( M )] M ) is R -sa-constructible.Proof. First, let us suppose that S ( M ) is a hypersurface.Locally on X , there exists a finite sequence of complex blow-up maps π : X ′ → X such that, denoting Z ′ := π − ( S ( M )), π | X ′ \ Z ′ is a biholomor-phism and Z ′ is a normal crossings hypersurface. By Proposition 1.3.8 (i) and Proposition 1.2.4, we have that it is sufficient to prove the statementfor M ∈ D bh ( D X ) such that S ( M ) is a normal crossings hypersurface and R Γ [ X \ S ( M )] M ≃ M .Now, as M is a bounded complex, by using inductively the distinguishedtriangles (1 . .
3) or (1 . .
4) of [16], one checks easily that it is sufficient to provethe statement for
M ∈
Mod h ( D X ) such that S ( M ) is a normal crossingshypersurface and R Γ [ X \ S ( M )] M ≃ M .Now, M can be considered as a meromorphic connection. By Theo-rem 1.4.3, locally on X , there exists a composition of modifications andramification maps π : X ′ → X such that D π ∗ M admits an elementary A -decomposition. Hence, by Proposition 1.3.8 (i) and (ii) it is sufficientto prove the statement for M ∈
Mod h ( D X ) such that S ( M ) is a normalcrossings hypersurface, R Γ [ X \ S ( M )] M ≃ M and it admits an elementary A -decomposition.This last case follows by Lemma 2.1.2.To conclude the proof of Proposition 2.1.3, let us consider the case of S ( M )an analytic set.2 Locally, there exist hypersurfaces H , . . . , H r such that S ( M ) ≃ H ∩ . . . ∩ H r . For sake of shortness, set V j := X \ H j , hence X \ S ( M ) ≃ V ∪ . . . ∪ V r .Let us prove the R -sa-constructibility of DR tX ( R Γ [ V ∪ ... ∪ V r ] M ) by inductionon r . The case r = 1 has been treated above.For r >
1, we have the following distinguished triangle,(2.3) R Γ [ V ∪ ... ∪ V r ] M → R Γ [ V ∪ ... ∪ V r − ] M⊕ R Γ [ V r ] M → R Γ [( V ∩ V r ) ∪ ... ∪ ( V r − ∩ V r )] M +1 → . By the inductive hypothesis, the second and the third terms of (2.3)have R -sa-constructible tempered De Rham complexes. It follows thatDR tX ( R Γ [ X \ S ( M )] M ) is R -sa-constructible. R Γ [ S ( M )] M In this subsection we are going to prove the following
Proposition 2.2.1.
Let X be an analytic manifold, M ∈ D bh ( D X ) . Then DR tX ( R Γ [ S ( M )] M ) is R -sa-constructible.Proof. Since the problem is local, we can add (or neglect when necessary)the additional hypothesis that the singular locus of M is connected.For sake of shortness, set Z := S ( M ).Let us prove our result with an induction on d := dim Z .Clearly, for any complex manifold Y and any N ∈ D bh ( D Y ) such thatdim S ( N ) = 0, DR tY R Γ [ S ( N )] N is R -sa-constructible.Now, suppose d := dim Z > Y and any N ∈ D bh ( D Y ) such that dim S ( N ) < d , DR tY R Γ [ S ( N )] N is R -sa-constructible. By Proposition 2.1.3, it follows that DR tY ( N ) is R -sa-constructible.We want to prove that, for any G ∈ D b R − c ( C X )(2.4) R H om C X ( G, DR tX ( R Γ [ Z ] M )) ∈ D b R − c ( C X ) . As the condition is local on X and as D b R − c ( C X ) is generated by objects ofthe form C U , for U ∈ Op( X sa ), it is sufficient to prove (2.4) with G ≃ C U , U ∈ Op c ( X sa ).First, we have the following sequence of isomorphisms in D b ( C X ) R H om C X ( C U , Ω tX L ⊗ ̺ ! D X ̺ ! R Γ [ Z ] M ) ≃ .2 The case of R Γ [ S ( M )] M ≃ R H om C X ( C U , Ω tX L ⊗ ̺ ! D X ̺ ! ( R Γ [ Z ] O X D ⊗ R Γ [ Z ] M )) ≃ R H om C X ( C U ⊗ C Z , Ω tX L ⊗ ̺ ! D X ̺ ! R Γ [ Z ] M ) ≃ R H om C X ( C U ∩ Z , Ω tX L ⊗ ̺ ! D X ̺ ! R Γ [ Z ] M ) . We used Theorem 1.2.3 (ii) and the fact that R Γ [ Z ] R Γ [ Z ] M ≃ R Γ [ Z ] M inthe first isomorphism and Lemma 1.3.6 with R := R Γ [ Z ] O X in the secondisomorphism.It follows that it is sufficient to prove (2.4) with G ≃ C U ∩ Z for all U ∈ Op c ( X sa ).Now, we are going to use some results of analytic geometry concerninganalytic sets, their regular and singular parts and their strict transform (see[9] or [2]); for the convenience of the reader we will briefly recall them here.Given an analytic set Z ⊂ X , a point z ∈ Z is said regular if there exists aneighborhood W of z such that Z ∩ W is a closed submanifold of W . It iswell known that the set of regular points of an analytic set Z ⊂ X , denoted Z r , is a dense open subset of Z . Furthermore Z s := Z \ Z r is a closedanalytic set called the singular part of Z whose dimension is smaller thatthe dimension of Z . Moreover, there exists a proper morphism of analyticmanifolds π : X ′ → X and a closed submanifold Z ′ of X ′ , called the stricttransform of Z , such that(2.5) π | Z ′ \ π − ( Z s ) : Z ′ \ π − ( Z s ) −→ Z \ Z s is an isomorphism.Now, it is sufficient to prove (2.4) with G ≃ C U ∩ Z r and G ≃ C U ∩ Z s for U ∈ Op c ( X sa ). Let us treat these two cases separately. The case of C U ∩ Z r . Let π : X ′ → X be the map described above in (2.5) and let Z ′ be the stricttransform of Z , set π Z ′ := π | Z ′ . Clearly, V := π − ( U ∩ Z r ) is a relativelycompact subanalytic subset of Z ′ such that C U ∩ Z r ≃ Rπ Z ′ ! C V .We have the following sequence of isomorphisms R H om C X ( C U ∩ Z r , Ω tX L ⊗ ̺ ! D X ̺ ! R Γ [ Z ] M ) ≃ (2.6) ≃ R H om C X ( Rπ Z ′ ! C V , Ω tX L ⊗ ̺ ! D X ̺ ! R Γ [ Z ] M ) ≃ Rπ ∗ R H om C Z ′ ( C V , π Z ′ ! (Ω tX L ⊗ ̺ ! D X ̺ ! R Γ [ Z ] M )) ≃ Rπ ∗ R H om C Z ′ ( C V , Ω tZ ′ L ⊗ ̺ ! D Z ′ ̺ ! ( D π Z ′ ∗ R Γ [ Z ] M )) . References
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Giovanni MorandoDipartimento di Matematica Pura ed Applicata,Universit`a degli Studi di Padova,Via Trieste 63, 35121 Padova, Italy.
E-mail address: [email protected]
Lehrstuhl f¨ur Algebra und ZahlentheorieUniversit¨atsstraße 1486159 Augsburg, Germany