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Two finiteness theorem for (a,b)-modules. Daniel Barlet∗. 16/01/08 8 0 0 2 n a Summary. J 8 2 We prove the following two results ] G 1. For a proper holomorphic function f : X → D of a complex manifold X A on a disc such that {df = 0} ⊂ f−1(0), we construct, in a functorial way, . for each integer p, a geometric (a,b)-module Ep associated to the (filtered) h t Gauss-Manin connexion of f. a m This first theorem is an existence/finiteness result which shows that geometric [ (a,b)-modules may be used in global situations. 1 2. For any regular (a,b)-module E we give an integer N(E), explicitely given v 0 from simple invariants of E, such that the isomorphism class of E bN(E).E 2 determines the isomorphism class of E. (cid:14) 3 4 This second result allows to cut asymptotic expansions (in powers of b) of . 1 elements of E without loosing any information. 0 8 0 AMS Classification : 32-S-05, 32S 20, 32-S-25, 32-S-40. : v i X Key words : (a,b)-module or Brieskorn modules, Gauss-Manin connexion, vanishing r cycles. a ∗ Barlet Daniel, Institut Elie Cartan UMR 7502 Nancy-Universit´e, CNRS, INRIA et Institut Universitaire de France, BP 239 - F - 54506 Vandoeuvre-l`es-Nancy Cedex.France. e-mail : [email protected] 1 Contents 1 Introduction. 3 2 The existence theorem. 4 2.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 A˜−structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 The existence theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Basic properties. 13 3.1 Definition and examples. . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 The regularity order. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3 Duality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.4 Width of a regular (a,b)-module. . . . . . . . . . . . . . . . . . . . . 27 4 Finite determination of regular (a,b)-modules. 31 4.1 Some more preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Finite determination for a rank one extension. . . . . . . . . . . . . . 32 4.3 The theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5 Bibliography 39 2 1 Introduction. The following situation is frequently met : we consider a vector space E of multi- valued holomorphic functions (possibly with values in a complex finite dimensional vector space V) with finite determination on a punctured disc around 0 in C, stable by multiplication by the variable z and by ”primitive”. These functions are determined by there formal asymptotic expansions at 0 of the type (Logz)j c (z).zα. α,j j! X (α,j)∈A×[0,n] where n is a fixed integer, where A is a finite set of complex numbers whose real parts are, for instance, in the interval ]−1,0], and where the c are in C[[z]]⊗V. α,j Define e(α,j) = zα.(Logz)j/j! and E(α,n) := ⊕n (C[[z]]⊗V).e(α,j). Then for j=0 each α, E(α,n) is a free C[[z]]−module of rank (n+ 1).dimV which is stable z by b := . To be precise, b is defined by induction on j ≥ 0 by the ”obvious” 0 formulasR: e(α+1,0) b[e(α,0)] = and for j ≥ 1 α+1 e(α+1,j) 1 b[e(α,j)] = − .b[e(α,j −1)]. α+1 α+1 So we have an inclusion E ⊂ ⊕ E(α) which is compatible with a := ×z and α∈A by b. Note that each E(α) is also a free finite rank C[[b]]−module which is stable by a. Let us assume now that E is also a C[[b]]−module which is stable by a. Then E has to be free and of finite rank over C[[b]]. Our aim in this situation is to under- stand and to describe the relations between the coefficients c of the asymptotic α,j expansions of elements in E. This leads to construct ”invariants” associated to the given E. As in the ”geometric” situations we consider the complex numbers α ∈ A are rationnal numbers, a change of variable of type t := z1/N, with N ∈ N∗, allows to reduce the situation to the case where all α are 0. Then one can use the ”nilpotent” operator d and the nilpotent part of the monodromy e → e dt α,j α,j−1 to construct some filtrations in order, in the good cases, to build a Mixte Hodge structure on a finite dimensional vector space associated to E. For instance, this is the case in A.N. Varchenkho’s description[V. 80] of the Mixte Hodge structure built by J. Steenbrink [St. 76] on the cohomology of the Milnor fiber of an holomorphic function with an isolated singularity at the origin of Cn+1. But it is clear that we loose some information in this procedure. The point of view which is to consider E itself as a left module on the (non commutative) algebra A˜:= { P (a).bν} ν X ν≥0 3 where the P are polynomials with complex coefficients is richer. This is evidenced ν by M. Saito result [Sa. 91]. The aim of the first part of this article is to build in a natural way, for any proper holomorphic function f : X → D of a complex manifold X, assumed to be smooth outside its 0−fiber X := f−1(0), a regular (geometric) (a,b)-module for each de- 0 gree p ≥ 0, which represent a filtered version of the Gauss-Manin connexion of f at the origin. This result is in fact a finiteness theorem which is a first step to refine the limite Mixte Hodge structure in this situation. It is interesting to remark that no K¨ahler assumption is used in this construction of these geometric (a,b)-modules. This obviuosly shows that (a,b)-modules are basic objects and that they are impor- tant not only in the study of local singularities of holomorphic functions but more generally in complex geometry . So it is interesting to have some tools in order to compute them. This is precisely the aim of the second part of this paper. We prove a finiteness result which gives, for a regular (a,b)-module E, an integer N(E), bounded by simple numerical invariants of E, such that you may cut the asymptotic expansions (in powers of b) of elements of E without any lost of information on the structure of the (a,b)-module E. It is well known that the formal asymptotic expansions for solutions of a regular differential system always converge, and also that such an integer exists for any meromorphic connexion in one variable (see [M.91] proposi- tion 1.12). But it is important to have an effective bound for such an integer easely computable from simple invariants of the (a,b)-module structure of E. 2 The existence theorem. 2.1 Preliminaries. Hereweshallcompleteandprecisetheresultsofthesection2of[B.07]. Thesituation we shall consider is the following : let X be a connected complex manifold of dimension n + 1 and f : X → C a non constant holomorphic function such that {x ∈ X/ df = 0} ⊂ f−1(0). We introduce the following complexes of sheaves supported by X := f−1(0) 0 1. The formal completion ”in f” (Ωˆ•,d•) of the usual holomorphic de Rham complex of X. 2. The sub-complexes (Kˆ•,d•) and (Iˆ•,d•) of (Ωˆ•,d•) where the subsheaves Kˆp and Iˆp+1 are defined for each p ∈ N respectively as the kernel and the image of the map ∧df : Ωˆp → Ωˆp+1 given par exterior multiplication by df. We have the exact sequence 0 → (Kˆ•,d•) → (Ωˆ•,d•) → (Iˆ•,d•)[+1] → 0. (1) 4 Note that Kˆ0 and Iˆ0 are zero by definition. 3. The natural inclusions Iˆp ⊂ Kˆp for all p ≥ 0 are compatible with the differential d. This leads to an exact sequence of complexes 0 → (Iˆ•,d•) → (Kˆ•,d•) → ([Kˆ/Iˆ]•,d•) → 0. (2) 4. We have a natural inclusion f∗(Ωˆ1) ⊂ Kˆ1 ∩ Kerd, and this gives a sub- C complex (with zero differential) of (Kˆ•,d•). As in [B.07], we shall consider also the complex (K˜•,d•) quotient. So we have the exact sequence 0 → f∗(Ωˆ1) → (Kˆ•,d•) → (K˜•,d•) → 0. (3) C We do not make the assumption here that f = 0 is a reduced equation of X , 0 and we do not assume that n ≥ 2, so the cohomology sheaf in degree 1 of the complex (Kˆ•,d•), which is equal to Kˆ1∩Kerd does not co¨ıncide, in general, with f∗(Ωˆ1). So the complex (K˜•,d•) may have a non zero cohomology C sheaf in degree 1. Recall now that we have on the cohomology sheaves of the following complexes (Kˆ•,d•),(Iˆ•,d•),([Kˆ/Iˆ]•,d•) and f∗(Ωˆ1),(K˜•,d•) natural operations a and b C with the relation a.b−b.a = b2. They are defined in a na¨ıve way by a := ×f and b := ∧df ◦d−1. The definition of a makes sens obviously. Let me precise the definition of b first in the case of Hp(Kˆ•,d•) with p ≥ 2 : if x ∈ Kˆp ∩Kerd write x = dξ with ξ ∈ Ωˆp−1 and let b[x] := [df ∧ξ]. The reader will check easily that this makes sens. For p = 1 we shall choose ξ ∈ Ωˆ0 such that ξ = 0 on the smooth part of X 0 (set theoretically). This is possible because the condition df ∧dξ = 0 allows such a choice : near a smooth point of X we can choose coordinnates such f = xk 0 0 and the condition on ξ means independance of x ,··· ,x . Then ξ has to be (set 1 n theoretically) locally constant on X which is locally connected. So we may kill 0 the value of such a ξ along X . 0 The case of the complex (Iˆ•,d•) will be reduced to the previous one using the next lemma. Lemme 2.1.1 For each p ≥ 0 there is a natural injective map ˜b : Hp(Kˆ•,d•) → Hp(Iˆ•,d•) ˜ ˜ which satisfies the relation a.b = b.(a+b). For p 6= 1 this map is bijective. 5 Proof. Let x ∈ Kˆp ∩Kerd and write x = dξ where ξ ∈ Ωˆp−1 (with ξ = 0 on X if p = 1), and set ˜b([x]) := [df ∧ξ] ∈ Hp(Iˆ•,d•). This is independant on 0 the choice of ξ because, for p ≥ 2, adding dη to ξ does not modify the result as [df ∧dη] = 0. For p = 1 remark that our choice of ξ is unique. This is also independant of the the choice of x in [x] ∈ Hp(Kˆ•,d•) because adding θ ∈ Kˆp−1 to ξ does not change df ∧ξ. Assume ˜b([x]) = 0 in Hp(Iˆ•,d•); this means that we may find α ∈ Ωˆp−2 such df ∧ξ = df ∧dα. But then, ξ −dα lies in Kˆp−1 and x = d(ξ −dα) shows that ˜ [x] = 0. So b is injective. Assume now p ≥ 2. If df ∧η is in Iˆp∩Kerd, then df∧dη = 0 and y := dη lies in Kˆp∩Kerd and defines a class [y] ∈ Hp(Kˆ•,d•) whose image by ˜b is [df ∧η]. ˜ This shows the surjectivity of b for p ≥ 2. ˜ For p = 1 the map b is not surjective (see the remark below). ˜ To finish the proof let us to compute b(a[x]+b[x]). Writing again x = dξ, we get a[x]+b[x] = [f.dξ +df ∧ξ] = [d(f.ξ)] and so ˜ ˜ b(a[x]+b[x]) = [df ∧f.ξ] = a.b([x]) (cid:4) which concludes the proof. Denote by i : (Iˆ•,d•) → (Kˆ•,d•) the natural inclusion and define the action of b on Hp(Iˆ•,d•) by b := ˜b ◦ Hp(i). As i is a−linear, we deduce the relation a.b−b.a = b2 on Hp(Iˆ•,d•) from the relation of the previous lemma. The action of a on the complex ([Kˆ/Iˆ]•,d•) is obvious and the action of b is zero. The action of a and b on f∗(Ωˆ1) ≃ E ⊗C are the obvious one, where E is C 1 X0 1 the rank 1 (a,b)-module with generator e satisfying a.e = b.e (or, if you prefer, 1 1 1 E := C[[z]] with a := ×z, b := z and e := 1). 1 0 1 Remark that the natural inclusionR f∗(Ωˆ1) ֒→ (Kˆ•,d•) is compatible with the C actions of a and b. The actions of a and b on H1(K˜•,d•) are simply induced by the corresponding actions on H1(Kˆ•,d•). Remark. The exact sequence of complexes (1) induces for any p ≥ 2 a bijection ∂p : Hp(Iˆ•,d•) → Hp(Kˆ•,d•) and a short exact sequence 0 → C → H1(Iˆ•,d•) →∂1 H1(Kˆ•,d•) → 0 (@) X0 because of the de Rham lemma. Let us check that for p ≥ 2 we have ∂p = (˜b)−1 and that for p = 1 we have ∂1 ◦ ˜b = Id. If x = dξ ∈ Kˆp ∩ Kerd then ˜b([x]) = [df ∧ξ] and ∂p[df ∧ξ] = [dξ]. So ∂p ◦˜b = Id ∀p ≥ 0. For p ≥ 2 and 6 df∧α ∈ Iˆp∩Kerd we have ∂p[df∧α] = [dα] and ˜b[dα] = [df∧α], so ˜b◦∂p = Id. For p = 1 we have ˜b[dα] = [df ∧(α−α )] where α ∈ C is such that α = α . 0 0 |X0 0 ˜ This shows that in degree 1 b gives a canonical splitting of the exact sequence (@). ˜ 2.2 A−structures. Let us consider now the C−algebra A˜:= { P (a).bν} ν X ν≥0 where P ∈ C[z], and the commutation relation a.b−b.a = b2, assuming that left ν and right multiplications by a are continuous for the b−adic topology of A˜. Define the following complexes of sheaves of left A˜−modules on X : (Ω′•[[b]],D•) and (Ω′′•[[b]],D•) where (4) +∞ Ω′•[[b]] := bj.ω with ω ∈ Kˆp j 0 X j=0 +∞ Ω′′•[[b]] := bj.ω with ω ∈ Iˆp j 0 X j=0 +∞ +∞ D( bj.ω ) = bj.(dω −df ∧ω ) j j j+1 X X j=0 j=0 +∞ +∞ a. bj.ω = bj.(f.ω +(j −1).ω ) with the convention ω = 0 j j j−1 −1 X X j=0 j=0 +∞ +∞ b. bj.ω = bj.ω j j−1 X X j=0 j=1 It is easy to check that D is A˜−linear and that D2 = 0. We have a natural ˜ inclusion of complexes of left A−modules ˜i : (Ω′′•[[b]],D•) → (Ω′•[[b]],D•). Remark that we have natural morphisms of complexes u : (Iˆ•,d•) → (Ω′′•[[b]],D•) v : (Kˆ•,d•) → (Ω′•[[b]],D•) and that these morphisms are compatible with i. More precisely, this means that we have the commutative diagram of complexes (Iˆ•,d•) u // (Ω′′•[[b]],D•) i ˜i (cid:15)(cid:15) (cid:15)(cid:15) (Kˆ•,d•) v // (Ω′•[[b]],D•) 7 The following theorem is a variant of theorem 2.2.1. of [B. 07]. Th´eor`eme 2.2.1 Let X be a connected complex manifold of dimension n + 1 and f : X → C a non constant holomorphic function with the following condition: {x ∈ X/ df = 0} ⊂ f−1(0). Thenthe morphismsofcomplexes u and v introducedaboveare quasi-isomorphisms. Moreover, the isomorphims that they induce on the cohomology sheaves of these com- plexes are compatible with the actions of a and b. This theorem builds a natural structure of left A˜−modules on each of the complex (Kˆ•,d•),(Iˆ•,d•),([Kˆ/Iˆ]•,d•) and f∗(Ωˆ1),(K˜•,d•) in the derived category of C bounded complexes of sheaves of C−vector spaces on X. Moreover the short exact sequences 0 → (Iˆ•,d•) → (Kˆ•,d•) → ([Kˆ/Iˆ]•,d•) → 0 (2) 0 → f∗(Ωˆ1) → (Kˆ•,d•),(Iˆ•,d•) → (K˜•,d•) → 0. (3) C ˜ are equivalent to short exact sequences of complexes of left A−modules in the derived category. Proof. We have to prove that for any p ≥ 0 the maps Hp(u) and Hp(v) are bijective and compatible with the actions of a and b. The case of Hp(v) is handled (at least for n ≥ 2 and f reduced) in prop. 2.3.1. of [B.07]. To seek completness and for the convenience of the reader we shall treat here the case of Hp(u). First we shall prove the injectivity of Hp(u). Let α = df ∧ β ∈ Iˆp ∩ Kerd and assume that we can find U = +∞bj.u ∈ Ω′′p−1[[b]] with α = DU. Then we j=0 j have the following relations P u = df ∧ζ, α = du −df ∧u and du = df ∧u ∀j ≥ 1. 0 0 1 j j+1 For j ≥ 1 we have [du ] = b[du ] in Hp(Kˆ•,d•); using corollary 2.2. of [B.07] j j+1 which gives the b−separation of Hp(Kˆ•,d•), this implies [du ] = 0,∀j ≥ 1 in j Hp(Kˆ•,d•). For instance we can find β ∈ Kˆp−1 such that du = dβ . Now, by de 1 1 1 Rham, we can write u = β +dξ for p ≥ 2, where ξ ∈ Ωˆp−2. Then we conclude 1 1 1 1 that α = −df ∧d(ξ +ζ) and [α] = 0 in Hp(Iˆ•,d•). 1 For p = 1 we have u = 0, du = 0 so [α] = [−df ∧dξ ] = 0 in H1(Iˆ•,d•). 0 1 1 We shall show now that the image of Hp(u) is dense in Hp(Ω′′•[[b]],D•) for its b−adic topology. Let Ω := +∞ bj.ω ∈ Ω′′p[[b]] such that DΩ = 0. The following j=0 j relations holds dω = df ∧ωP ∀j ≥ 0 and ω ∈ Iˆp. The corollary 2.2. of [B.07] j j+1 0 again allows to find β ∈ Kˆp−1 for any j ≥ 0 such that dω = dβ . Fix N ∈ N∗. j j j We have N D( bj.ω ) = bN.dω = D(bN.β ) j N N X j=0 8 and Ω := N bj.ω − bN.β is D−closed and in Ω′′p[[b]]. As Ω − Ω lies N j=0 j N N in bN.Hp(Ω′′P•[[b]],D•), the sequence (Ω ) converges to Ω in Hp(Ω′′•[[b]],D•) N N≥1 for its b−adic topology. Let us show that each Ω is in the image of Hp(u). N Write Ω := N bj.w . The condition DΩ = 0 implies dw = 0 and N j=0 j N N dw = df ∧wP= 0. If we write w = dv we obtain d(w +df ∧v ) = 0 N−1 N N N N−1 N and Ω − D(bN.v ) is of degree N − 1 in b. For N = 1 we are left with N N w + b.w − (−df ∧ v + b.dv ) = w + df ∧ v which is in Iˆp ∩ Kerd because 0 1 1 1 0 1 dw = df ∧dv . 0 1 To conclude it is enough to know the following two facts i) The fact that Hp(Iˆ•,d•) is complete for its b−adic topology. ii) The fact that Im(Hp(u))∩bN.Hp(Ω′′•[[b]],D•) ⊂ Im(Hp(u)◦bN) ∀N ≥ 1. Let us first conclude the proof of the surjectivity of Hp(u) assuming i) and ii). For any [Ω] ∈ Hp(Ω′′•[[b]],D•) we know that there exists a sequence (α ) N N≥1 in Hp(Iˆ•,d•) with Ω − Hp(u)(α ) ∈ bN.Hp(Ω′′•[[b]],D•). Now the property ii) N implies that we may choose the sequence (α ) such that [α ]−[α ] lies N N≥1 N+1 N in bN.Hp(Iˆ•,d•). So the property i) implies that the Cauchy sequence ([α ]) N N≥1 converges to [α] ∈ Hp(Iˆ•,d•). Then the continuity of Hp(u) for the b−adic topologies coming from its b−linearity, implies Hp(u)([α]) = [Ω]. The compatibility with a and b of the maps Hp(u) and Hp(v) is an easy exercice. Let us now prove properties i) and ii). The property i) is a direct consequence of the completion of Hp(Kˆ•,d•) for its b−adic topology given by the corollary 2.2. of [B.07] and the b−linear isomorphism ˜b between Hp(Kˆ•,d•) and Hp(Iˆ•,d•) constructed in the lemma 2.1.1. above. To prove ii) let α ∈ Iˆp ∩Kerd and N ≥ 1 such that α = bN.Ω+DU where Ω ∈ Ω′′p[[b]] satisfies DΩ = 0 and where U ∈ Ω′′p−1[[b]]. With obvious notations we have α = du −df ∧u 0 1 ··· 0 = du −df ∧u ∀j ∈ [1,N −1] j j+1 ··· 0 = ω +du −df ∧u 0 N N+1 which implies D(u + b.u + ···+ bN.u ) = α + bN.du and the fact that du 0 1 N N N lies in Iˆp ∩Kerd. So we conclude that [α]+bN.[du ] is in the kernel of Hp(u) N which is 0. Then [α] ∈ bN.Hp(Iˆ•,d•). (cid:4) 9 Remark. The map β : (Ω′[[b]]•,D•) → (Ω′′[[b]]•,D•) defined by β(Ω) = b.Ω commutes to the differentials and with the action of b. It ˜ induces the isomorphism b of the lemma 2.1.1 on the cohomology sheaves. So it is a quasi-isomorphism of complexes of C[[b]]−modules. To prove this fact, it is enough to verify that the diagram (Kˆ•,d•) v // (Ω′[[b]]•,D•) ˜b β (cid:15)(cid:15) (cid:15)(cid:15) (Iˆ•,d•) u //(Ω′′[[b]]•,D•) induces commutative diagrams on the cohomology sheaves. But this isclear because if α = dξ lies in Kˆp∩Kerd wehave D(b.ξ) = b.dξ−df∧ξ so Hp(β)◦Hp(v)([α]) = Hp(u)◦Hp(˜b)([α]) in Hp(Ω′′[[b]]•,D•). (cid:4) 2.3 The existence theorem. Let us recall some basic definitions on the left modules over the algebra A˜. D´efinition 2.3.1 An (a,b)-module is a left A˜−module which is free and of finite rank on the commutative sub-algebra C[[b]] of A˜. An (a,b)-module E is 1. local when ∃N ∈ N such that aN.E ⊂ b.E; 2. simple pole when a.E ⊂ b.E; 3. regular when it is contained in a simple pole (a,b)-module; 4. geometric when it is contained in a simple pole (a,b)-module E♯ such that the minimal polynomial of the action of b−1.a on E♯ b.E♯ has its roots in Q+∗. (cid:14) We shall give more details and examples of (a,b)-modules in the section 3. Now let E be any left A˜−module, and define B(E) as the b−torsion of E, that is to say B(E) := {x ∈ E / ∃N bN.x = 0}. Define A(E) as the a−torsion of E and Aˆ(E) := {x ∈ E / C[[b]].x ⊂ A(E)}. Remark that B(E) and Aˆ(E) are sub-A˜−modules of E but that A(E) is not stable by b. ˜ D´efinition 2.3.2 A left A−module E is small when the following conditions hold 10

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