Integral representations for solutions of exponential Gauß-Manin systems

Bulletin

SOCIÉTÉ MATHÉMATIQUE DE FRANCE

Publié avec le concours du Centre national de la recherche scienti�que

de la SOCIÉTÉ MATHÉMATIQUE DE FRANCE

Tome 136 Fascicule 4

2008

pages 505-532

INTEGRAL REPRESENTATIONS FOR SOLUTIONS OF EXPONENTIAL

GAUSS-MANIN SYSTEMS

Marco Hien & Céline Roucairol

INTEGRAL REPRESENTATIONS FOR SOLUTIONS OF EXPONENTIAL GAUSS-MANIN SYSTEMS

by Marco Hien & Céline Roucairol

Abstract. — Letf, g :U → A¹ be two regular functions from the smooth affine complex varietyUto the affine line. The associated exponential Gauß-Manin systems on the affine line are defined to be the cohomology sheaves of the direct image of the exponential differential systemO_Ue^gwith respect tof. We prove that its holomorphic solutions admit representations in terms of period integrals over topological chains with possibly closed support and with rapid decay condition.

Résumé(Représentations intégrales des solutions des systèmes de Gauß-Manin expo- nentiels)

Soientf, g:U→A¹deux fonctions régulières sur une variété affine lisseUà valeurs dans la droite affine. On leurs associe des systèmes de Gauß-Manin sur la droite affine définis comme étant les faisceaux de cohomology de l’image directe par f du sys- tème différentiel exponentiel O_Ue^g. Nous prouvons que leurs solutions holomorphes admettent des représentations sous forme d’intégrales de périodes sur des chaînes to- pologiques à support éventuellement fermé avec une condition de décroissance rapide.

Texte reçu le 5 novembre 2007 et le 7 mars 2008, accepté le 7 juillet 2008

Marco Hien, NWF I – Mathematik, Universität Regensburg, 93040 Regensburg, Germany

• E-mail :marco.hien@mathematik.uni-regensburg.de

Céline Roucairol, Lehrstuhl für Mathematik VI, Universität Mannheim, A5 6, 68161 Mannheim, Germany • E-mail :celine.roucairol@uni-mannheim.de

2000 Mathematics Subject Classification. — 32S40, 32C38, 14F40.

Key words and phrases. — Gauß-Manin systems,D-modules.

BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE 0037-9484/2008/505/$ 5.00

©Société Mathématique de France

1. Introduction

Let U be a smooth affine complex variety of dimension n and let f, g : U →A¹ be two regular functions. We consider the flat algebraic connection

∇ : OU → Ω¹_U on the trivial line bundle OU defined as ∇u = du+dg·u.

Let us denote the associated holonomic DU-module by OUe^g, i.e. the trivial O_U-module on which any vector fieldξonU acts via the associated derivation

∇ξ induced by the connection. LetN be the direct imageN :=f+(OUe^g)in the theory of D-modules (see e.g. [2]). Then N is a complex ofD_A¹-modules and we consider itsk-th cohomology sheaf

H^kf+(OUe^g), a holonomicD_A¹-module.

There is an alternative point of view of H^kN in terms of flat connections:

outside a finite subset Σ₁ ⊂A¹, theO_A1-module H^kf₊(O_Ue^g)is locally free, hence a flat connection. This connections coincides with the Gauß-Manin con- nection on the relative de Rham cohomology

H^k(N)|_A1rΣ₁ ∼= R^k+n−1f_∗(Ω·

U/A¹,∇),∇_GM

|_A1rΣ₁,

i.e. the corresponding higher direct image of the complex of relative differential forms on U with respect to f and the differential induced by the absolute connection, as in [6]. It is a flat connection overA¹rΣ1.

In the case g = 0, which is classically called the Gauß-Manin system off, it is well-known that the solutions admit integral representations (see [10] and [12]). The aim of the present article is to give such a description of the solutions in the more general case of the Gauß-Manin system of the exponential module OUe^g, which we will call exponential Gauß-Manin system. A major difficulty lies in the definition of the integration involved. In the case of vanishingg, the connection is regular singular at infinity and the topological cycles over which the integrations are performed can be chosen with compact support inside the affine variety U. In the exponential case g 6= 0, the connection is irregular singular at infinity and we will have to consider integration paths approaching the irregular locus at infinity.

A systematic examination of period integrals in more general irregular sin- gular situations over complex surfaces has been carried out in [5] resulting in a perfect duality between the de Rham cohomology and some homology groups, the rapid decay homology groups, in terms of period integrals. We will prove in this article that this result generalizes to arbitrary dimension in case of an elementary exponential connection Oe^g as before, see Theorem2.7.

With this tool at hand, we construct a local system H^rd_k onA¹rΣ2, the stalk at point a t of which is the rapid decay homology H_k^rd(f⁻¹(t), e^−gt) of the restriction of O_Ue^−g to the fibref⁻¹(t). LetΣ = Σ₁∪Σ₂. We prove that

o

given a sectionct⊗e^gt in the local system H^rd_k+n−1 and a relative differential form ω in Ω^k+n−1_rel (f⁻¹(A¹rΣ)) (which describes the analytification of H^kN as in the obvious analytic analogue to Proposition 3.3), the integral

(1)

Z

ct

ω|_f⁻¹_(t)·e^gt

gives a (multi-valued) holomorphic solution of the exponential Gauß-Manin system. As a consequence of the duality proved in Theorem 2.7, we deduce that all (multi-valued) holomorphic solutions on A¹rΣ can be obtained by this construction, i.e. we achieve the following theorem (Theorem 3.5) which is the main result of this article:

Theorem 1.1. — For any simply connected open subsetV ⊂C rΣ, the space of holomorphic solutions of the exponential Gauß-Manin system H^kN is iso- morphic to the space of sections in the local system H^rd_k+n−1 overV.

The isomorphism in the theorem is given in terms of the integration (1). We want to remark that this theorem generalizes similar results by F. Pham for the Fourier-Laplace transform of the Gauß-Manin systems of a regular function h : U → A¹ (consider f : A¹×U → A¹ to be the first canonical projection andg:A¹×U →A¹ given byg(x, u) =xh(u), cf. [11]). It also includes other well-known examples as the integral representations of the Bessel-functions for instance (cf. the introduction of [1] and the final remark of this article).

2. The period pairing

Consider the situation described in the introduction, namely f, g:U →A¹ being two regular functions on the smooth affine complex varietyU. The main result of this work is to give a representation of the Gauß-Manin solutions in terms of period integrals. The proof relies on a duality statement between the algebraic de Rham cohomology of the exponential connection associated to g and some Betti homology groups with decay condition. The present section is devoted to the proof of this duality statement, which is a generalization of the analogous result for surfaces in [5] to the case of exponential line bundles in arbitrary dimensions.

2.1. Rapid decay homology. — Let us start with any smooth affine complex variety Ut and a regular function gt : Ut → A¹. Note, that the index t is irrelevant in this section but will become meaningful later in the application to the Gauß-Manin system where U_twill denote the smooth fibresf⁻¹(t).

We assume that Ut is embedded into a smooth projective variety Xt and thatgtextends to a meromorphic mappinggt:Xt→P¹. Note, that we do not impose any further conditions on the pole divisor D_t:=g⁻¹_t (∞).

Let us denote byO_Xt[∗D_t]the sheaf of meromorphic functions on X_t with poles alongDt. We will writeOe^gt for the flat meromorphic connection

∇:OX_t[∗Dt]−→Ω¹_X

t/C[∗Dt], u7→du+u·dgt

on the trivial meromorphic line bundleOXt[∗Dt].

In [5], a duality pairing between the de Rham cohomology of a flat meromor- phic connection with poles along a divisor with normal crossings (admitting a good formal structure) and a certain homology theory, the rapid decay homol- ogy, is constructed in the casedim(Xt) = 2, generalizing previous constructions of Bloch and Esnault for curves. For connections of exponential type lying in the main focus of this work, we will now generalized these results to the case of arbitrary dimension and also weakening the normal crossing assumption on the pole divisor. We remark that a crucial point in [5] is to work with agood compactification with respect to the given vector bundle and connection. For connections of the typeOe^gt as above, we can always pull-back to such a good situation by a finite blow-up process.

We now want to define the rapid decay homology of the dual connection Oe^−gt. Since we do not assume that Dtis a normal crossing divisor, we have to modify the definition of [5] for our present purposes. We start with the given meromorphic map g_t : X_t → P¹. Consider the real oriented blow-up π : Pf¹ → P¹ of the point ∞ in P¹ and let S_∞¹ := π⁻¹(∞) be the circle at infinity. We will use the following local presentation of the real oriented blow- up of∞ ∈P¹

π⁻¹(P¹r0)∼=R⁺0 ×S¹−→^π P¹r0, (r, e^iϑ)7→1 re^iϑ and identify S_∞¹ =π⁻¹(∞)with0×S¹. We define

(2) Xft=Xt×g_tfP¹

as the fibre product of Xt and fP¹ overP¹ and denote the associated maps as in the following diagram:

(3) Xft

π_X

‹ gt //

fP¹

π

X_t gt //P¹

o

Restricted to π⁻¹(Ut), the map πX is a homeomorphism to Ut by which we considerUtas an open subset ofXft. Lete:Ut,→Xftbe the embedding. Addi- tionally, letC^−pX‹t

denote the sheaf associated to the presheafV 7→S_p(fX_t,Xf_trV) of Q-vector spaces, whereSp(Y)denotes the groups of piecewise smooth sin- gular p-chains in Y and Sp(Y, A) the relative chains for a pair of topological spacesA⊂Y.

We adopt the standard sign convention for the resulting complex C⁻·

‹

X_t. i.e.

the differential of C⁻·

‹

X_t will be given by(−1)^r times the topological boundary operator ∂onC^X−r‹t

. Now, let

(4) N :=

ß

ϑ:=e^iϕ∈S_∞¹

3π

2 ≤ϕ≤ 5π 2

™

⊂S_∞¹ ⊂fP¹

denote the closed half-circle in S_∞¹ of non-rapid decay in the sense that the functionr7→exp(¹_rϑ)has rapid decay forr→0if and only ifϑ6∈ N.

Consider the subspaces

(5) R_t:=Xf_trg‹_t⁻¹N ⊂Xf_t and S_t:=‹g_t⁻¹S_∞¹ ∩ R_t⊂Xf_t. Let C⁻·

Rt,St be the complex of chains in Rt relative to St =RtrUt, i.e. the sheaf associated to the presheaf

V 7→S·(Rt,(RtrV)∪ St) onRt. Ifcdenotes ap-chainc∈Γ(V,C_R^−p

t,St)represented by a mapc: ∆^p → Rt

from the standardp-simplex∆^ptoRt– for simplicity assume thatcmaps the interior of ∆^p to U_t – then the function exp(g_t◦ c(z)) has rapid decay for z∈interior(∆^p)approaching the boundary∂∆^pby the definition ofRt.

Definition 2.1. — Consider the inclusionsUt

,→ Rj t

,→κ Xft,κbeing a closed embedding, and let E denote the trivial local system on Ut associated to the dual connectionOUte^−gt (spanned bye^gt). Therapid decay complexis defined to be the complex of sheaves

(6) C^rdX‹t

(e^−gt) :=κ! C⁻·

R_t,S_t⊗j∗E

onXf_t. Therapid decay homologyofOe^−gt is the corresponding hypercohomol- ogy

H_k^rd(Ut, e^−gt) :=H^−k(fXt,C^Xrd‹t

(e^−gt)).

We will see in Proposition 2.3 that the rapid decay homology is indeed independent of the choice of the compactification Xt such that gt admits a meromorphic continuationg_t:X_t→P¹.

The usual barycentric subdivision operator onC⁻·

R_t,S_t induces a subdivision operator on the rapid decay complex. The latter is therefore easily seen to be a homotopically fine complex of sheaves (cp. [18], p. 87). Hence, the rapid decay homology can be computed as the cohomology of the corresponding complex of global sections:

(7) H_k^rd(Ut, e^gt) =H^−k

Γ(fXt,C^Xrd‹t

(e^−gt)) .

2.2. The pairing. — We will now define the period pairing in the situation described above. SinceU_tis assumed to be affine, the de Rham cohomology of OU_te^gt can be computed on the level of global sections

H_dR^p (Ut, e^gt) :=H^p(Ut,(Ω·

Ut/C,∇t))∼=H^p(Γ(Ut,Ω·

Ut/C),∇t), with∇t(ω) =dω+dg_t∧ω for a local sectionω ofΩ^q_U

t/C.

Now, if we have a global rapid decay chainc⊗e^gt ∈Γ(fX_t,C^rd,−p

‹

Xt

(e^−gt))with respect to the dual bundle O_Ute^−gt and a meromorphic p-form ω, then the integralR

cωe^gt converges because the rapid decay ofe^gt alongcannihilates the moderate growth of the meromorphic ω. Letcτ denote the topological chain one gets by cutting off a small tubular neighborhood with radius τ around the boundary ∂∆^p from the given topological chain c. Then, for c⊗e^gt ∈ C^Xrd,p‹t

(e^−gt)(fX_t)and η ∈Ω^p−1(∗Dt) a meromorphic(p−1)-form, we have the

“limit Stokes formula”

Z

c

(∇tη)e^gt = lim

τ→0

Z

c_τ

(∇tη)e^gt = lim

τ→0

Z

∂cτ

η e^gt= Z

∂c−Dt

η e^gt

where in the last step we used that by the given growth/decay conditions the integral over the faces of∂cτ ’converging’ against the faces of ∂ccontained in Dtvanishes.

The limit Stokes formula easily shows in the standard way that integrating a closed differential form over a given rd-cycle (i.e. with vanishing boundary value) only depends on the de Rham class of the differential form and the rd-homology class of the cycle. Thus, we have:

Proposition 2.2. — Integration induces a well defined bilinear pairing (8) H_dR^p (U_t, e^gt)×H_p^rd(U_t, e^−gt)−→C, ([ω],[c⊗e^gt])7→

Z

c

ω e^gt,

which we call the period pairingof OUte^gt.

o

2.3. Independence of choice of a good compactification. — Assume that we have two compactifications Xt and X_t⁰ of Ut such thatgt: Ut →A¹ admits mero- morphic extensionsg_t:X_t→P¹as well ash_t:X_t⁰→P¹. Passing to a common compactification (e.g. the closure ofUt inXt×X_t⁰), we can assume that there is a map b : X_t⁰ → Xt compatible with ht and gt. If Xft and Xf_t⁰ denote the associated spaces as in (3), the compatibilityht=gt◦bproduces a lifteb

Xf_t⁰ eb //

πX⁰

Xft

πX

X⁰_t b //X_t,

such thatg‹t◦eb=h‹t.

Proposition 2.3. — In the situation above, we have H^−k(fX_t⁰,CX^rd‹_t⁰(e^−ht)) =H^−k(fXt,C^Xrd‹t

(e^−gt)).

Proof. — The proposition follows by standard arguments: We have to consider the subspaces of rapid decay, i.e.

R⁰_t:=Xf_t⁰reh⁻¹_t (N) ^κ

0

,→Xf_t⁰ andR_t:=Xf_treg_t⁻¹(N),→^κ Xf_t

as well as their ’boundaries’S_t⁰ =R⁰_trUt and St=RtrUt. Obviously, the liftebmapsR⁰_tto Rt, i.e. we have the diagram

Ut  o j⁰ //

j >

>>

>>

>>

> R⁰_t eb

 κ⁰ //Xf_t⁰

eb

Rt  κ //Xft,

Regarding the definition of the rapid decay complex (6), keeping in mind that the local systemE =e^−gtCU_t is trivial, we have to show that

H^−k(fX_t⁰, κ⁰_!C⁻·

R⁰_t,S_t⁰)∼=H^−k(fXt, κ!C⁻·

Rt,St).

The left hand side is equal toH^−k(fXt,Reb_∗κ⁰_!C⁻·

R⁰_t,S⁰_t). Sinceeb is proper, we know that

Reb_∗κ⁰_!C⁻·

R⁰_t,S⁰_t =κ_!Reb_∗C⁻·

R⁰_t,S_t⁰, so that it remains to prove that Reb_∗C⁻·

R⁰_t,S⁰_t = C⁻·

Rt,St. Since the complex of sheavesC⁻·

R⁰_t,S_t⁰ is homotopically fine, we can deduce thatReb_∗C⁻·

R⁰_t,S_t⁰ =eb_∗C⁻·

R⁰_t,S⁰_t. The latter complex is isomorphic to C⁻·

Rt,St. To see this, note that the local

sections c ∈ eb_∗C_R^−p0

t,S_t⁰ can be represented by formal linear combinations of maps ∆^p → R⁰ from the standard p-simplex to R⁰ which map the interior int(∆^p) to Ut. By continuity, these maps are determined by their restriction to the interior int(∆^p)of the standard simplex. Since h‹t =eb◦g‹t, the closure of c(int(∆^p))inXf_t⁰ is contained in R⁰_t if and only if the corresponding closure in Xf_t is contained in R_t. This determines the isomorphism betweeneb_∗C_R^−p0

t,S_t⁰

andC_R^−p

t,St.

2.4. Comparison with the definition in[5]. — In [5], rapid decay homology was defined for arbitrary meromorphic connections admitting a good formal struc- ture on surfaces in the sense of Sabbah ([17]). One of the main features is to start with a good compactification X_t with respect to the given connection∇ in the sense that ∇ admits a good formal decomposition (see loc. cit.). In particular, the locus at infinity Dt :=XtrUt will be a divisor with normal crossings. In our present situation, the simple type of the given connection OUe^g allows to omit the normal crossing condition and additionally to gener- alize to arbitrary dimension. We now want to prove that the two definitions, namely Definition 2.1 and the one in [5] coincide. This comparison will be important in the proof of the duality of the period pairing in Theorem2.7.

Let us first recall the definition of [5]. Let gt:Xt→P¹ be a meromorphic extension of gt : Ut → A¹ and assume that Dt := XtrUt is a divisor with normal crossings. Consequently, we can consider the real oriented blow-up b:X_t→X_tof the components ofD_t(cp. [8]). Locally at a pointx⁰∈D_t, we can choose coordinates x1, . . . , xn such that locallyDt={x1· · ·xk = 0} with some 1≤k≤n. In these coordinates, breads as

((rν, e^iϑν)^k_ν=1, xk+1, . . . , xn)7→(r1e^iϑ1, . . . , rke^iϑk, xk+1, . . . , xn) withrν ∈R⁺0 ande^iϑν ∈S¹.

Similarly to the construction above, let C⁻·

Xt,Dt be the complex of chains relative to the boundary D_t:=b⁻¹(D_t), i.e. the sheaf associated to

V 7→S·(X_t,(X_trV)∪D_t).

Restricted to U_t, b is an isomorphism and hence we can consider U_t as a subspace of X_t. Letj:U_t,→X_tdenote the inclusion.

Let c⊗e^gt ∈ Γ(V,C⁻·

X_t,D_t ⊗j_∗E) be a local section over some open V ⊂ Xt, where we recall that E denotes the local system associated to the dual connection Oe^−gt. Assume that c is represented by a piecewise smooth map from the standard p-simplex ∆^p toX_t.

Now, consider a pointy∈c(∆^p)∩Dt∩V. Choose local coordinatesx1, . . . , xd

ofX_taroundy= 0such thatD_t={x₁· · ·x_k = 0}.

o

Definition 2.4. — We call c⊗e^gt a rapid decay chain on Xt, if for all y as above, the function e^gt(x) has rapid decay for the argument approaching Dt, i.e. if for all N ∈N^k there is aC_N >0 such that

|e^gt(x)| ≤C_N· |x₁|^N1· · · |x_k|^Nk for allx∈(c(∆^p)rD_t)∩V with small |x₁|, . . . ,|x_k|.

In case V ∩Dt = ∅, we do not impose any condition on c⊗e^gt. The subcomplex of C⁻·

X_t,D_t ⊗j_∗E consisting of all rapid decay complexes will be denoted by C_X^rd

t.

In this situation withDta normal crossing divisor, we can now compare the complexC_X^rd_t on the real oriented blow-upXtand the complexC^Xrd‹t

on the space Xf_t=X_t×_gtPf¹ introduced in (2), withDf_t:=Xf_trU_t. We first observe that there is a canonical lift ofg

Xt

g //

b

fP¹

π

X_t g //P¹

which can be described as follows using the local description above.

Consider a point y ∈ Dt and choose coordinates as above such that locally Dt = {x1· · ·xk = 0} and y = (0, . . . ,0, xk+1, . . . , xn). Let ξ := ((0, e^iζν)^k_ν=1, x_k+1, . . . , x_n) be an arbitrary point in b⁻¹(y). Then g(ξ) is the argument of g(x) forx∈ Ut approaching y in the direction given by(ζ1, . . . , ζk):

g(ξ) = lim

r→0argg(re^iζ1, . . . , re^iζk, xk+1, . . . , xn)∈S¹=π⁻¹(∞).

Letg:X_t→Xf_tdenote the induced mapping also. We now prove the following proposition.

Proposition 2.5. — In the situation above, we have Rg∗C_X^rd_t =C^Xrd‹t

.

Proof. — The argument is similar to the one in Proposition 2.3. Since both complexes are homotopically fine, we can applyg_∗directly toC_X^rd

t. LetV ⊂Xft

be some open subset with V ∩Df_t 6= 0. Now, a piecewise smooth p-chain c : ∆^p → Xt which maps the interior int(∆^p) to Ut induces an element of g∗C^rd_Xt(V)if and only if all directionsξ in the closurec(∆^p)ofc(int(∆^p))⊂Xt

are rapid decay directions forg, in other words if and only if the implication ξ∈c(∂∆^p)⇒g(ξ)∈ R_t

with the notationRtfrom (5) holds. But this is equivalent to saying thatc⊗e^gt defines an element ofC^Xrd‹t

. Thereforeg∗C^rd_Xt =C^Xrd‹t

.

Corollary 2.6. — The definitions of rapid decay homology in Definition2.1 and the one in [5], Definition 2.3-4, coincide.

Proof. — By Proposition 2.3, the rapid decay homology as in Definition 2.1 is independent of the choice of compactification with meromorphic extension of g_t. Hence, we can choose X_t with normal crossing complement and the assertion follows from Proposition 2.5.

2.5. Duality. — We will now prove the following duality theorem for the con- nection OUte^gt in arbitrary dimensiondimU_t.

Theorem 2.7. — The period pairing (8)is a perfect pairing of finite dimen- sional vector spaces.

The proof given in the following is a generalization of the one in [5] for the special case of a connection O_Ute^gt to arbitrary dimensiondim(X_t)∈N. Proof. — Using resolution of singularities, we can find a compactification Xt

ofUtsuch that

1. Dt:=XtrUtis a divisor with normal crossings and 2. g_t:U_t→A¹admits a meromorphic extension to X_t.

Let us denote the extension again by gt:Xt→P¹. Due to Corollary2.6, the rapid decay homology can be computed by the rapid decay complexC_X^rd

t on the real oriented blow-up b: X_t →X_t of the local irreducible components of D_t, i.e. H_k^rd(Ut, e^−gt) =H^−k(Xt,C_X^rd

t).

The proof will rely on a local duality on Xt. To prepare for it, we will describe the period pairing as a local pairing of complexes of sheaves on X_t. We recall the corresponding notions of [5].

Let A^modD_X ^t

t denote the sheaf of functions onXt which are holomorphic on U_t^an⊂Xtand of moderate growth alongDt:=b⁻¹(Dt)and letA^<D_X ^t

t denote the subsheaf of such functions that are infinitely flat alongDt, i.e. all of whose partial derivatives of arbitrary order vanish on D_t.

We call DR^modD_X ^t

t (e^gt) :=A^modD_X ^t

t ⊗_b−1(O_Xt)b⁻¹(DRX_t^an(OX_t[∗Dt]e^gt)) themoderate de Rham complexofO_Ute^gt and

DR^<D_X ^t

t (e^gt) :=A^<D_X ^t

t ⊗_b−1(O_Xt)b⁻¹(DRX^an_t (OX_t[∗Dt]e^gt))

o

the asymptotically flat de Rham complex. Note that the first complex com- putes the meromorphic de Rham cohomology of the meromorphic connection O_Xt[∗D_t]e^gt (cp. [17], Corollaire 1.1.8).

Now, the wedge product obviously defines a pairing of complexes of sheaves onXt:

(9) DR^modD_X ^t

t (e^gt)⊗_CDR^<D_X ^t

t (e^−gt)→DR^<D_X ^t

t (OX_t, d),

where the right hand side denotes the asymptotically flat de Rham complex of the trivial connection. We will compare this pairing to the periods pairing from above.

Before that, let us introduce the sheafDb^rd,−s_X

t of rapid decay distributions onXt, the local section of which are distributions

ϕ∈Db^−s_X

t(V) := Homcont(Γc(V,Ω^∞,s_X

t ),C) on the spaceΩ^∞,s_X

t ofC^∞differential forms onXtof degreeswith compact sup- port inX_tsatisfying the following condition: we choose coordinatesx₁, . . . , x_n on Xt such that locally onV one hasDt={x1· · ·xk = 0}. Then we require that for any compact K ⊂V and any elementN ∈N^k there are m∈N and C_K,N >0such that for any test formηwith compact support inKthe estimate (10) |ϕ(η)| ≤CK,N

X

i

sup

|α|≤m

sup

K

{|x|^N|∂^αfi|}

holds, whereαruns over all multi-indices of degree less than or equal tomand

∂^α denotes the α-fold partial derivative of the coefficient functions fi of η in the chosen coordinates.

Integration along chains induces a pairing of complexes of sheaves (11) DR^modD_X ^t,s

t (e^gt)⊗ C_X^rd,−r

t (e^−gt)→Db^rd,s−r_X

t

to be defined as follows: for any local s-form ω of DR^modD_X ^t

t (e^gt), any rapid decay chain c⊗e^gt ∈ Γ(V,C_X^rd,−r

t (e^−gt)) and any test form η ∈ Γ_c(V,Ω^∞,p_X

t ) withp=r−s, the decay and growth assumptions ensure that the integral (12)

Z

c

η∧(e^gt·ω)

converges and moreover, that the distribution that mapsη to (12) satisfies the condition (10) from above.

Note that Db^rd,−·

X_t is a fine resolution of e!CU_t^an[2d] where e : U_t^an ,→ Xt

denotes the inclusion and d:= dim_C(Xt). It follows that H⁰(Xt,Db^rd,·

X_t )∼=C in a standard way. The resulting pairing H_dR^p (Ut, e^gt)×H_p^rd(Ut, e^−gt) → C induced by (11) on cohomology in degree zero coincides with the period pairing (Proposition 2.2).

The proof of Theorem 2.7 now splits into the following intermediate steps (cp. with [5]):

Claim 1. — The complexesDR^modD_X ^t

t (e^gt)andDR^<D_X ^t

t (e^−gt)have cohomology in degree zero only.

Proof. — In both cases the argument relies on an existence result of a certain linear partial differential system with moderate or rapidly decaying coefficients.

To be more precise, consider the local situation at some point x0 ∈ Dt = {x1· · ·xk = 0}and letϑ∈b⁻¹(x0)'(S¹)^k be a direction inDtoverx0. Then the complex of stalks atϑwhich we have to consider is given as

(13)

· · · −→

A^?D_X ^t

t ⊗_b⁻¹_OXtb⁻¹Ω^p_X

t

ϑ

∇_t

−→

A^?D_X ^t

t ⊗_b⁻¹_OXtb⁻¹Ω^p+1_X

t

ϑ−→ · · ·, where ? stands for either<or mod. If we describe Ω^p_X

t in terms of the local basis dx_I for all I ={1 ≤i₁ <· · · < i_p ≤d}, the covariant derivation ∇_t in degreepreads as

X

#I=p

wIdxI 7→ X

#J=p+1

X

j∈J

sgnJ(j)(Qjw_Jr{j}) dxJ

where

Qju:= ∂

∂xj

u± ∂gt

∂xj

·u,

with the negative sign in case of e^gt and the positive in case ofe^−gt. Also, we letsgnJ(j) = (−1)^ν forJ ={j1<· · ·< jp+1}andjν=j.

Now, letω be a germ of a section ofA^?D_X ^t

t ⊗b⁻¹Ω^p+1_X

t written in the chosen coordinates as

ω= X

#J=p+1

wJdxJ,

such that ∇tω = 0. We have to find a p-form η with appropriate growth condition such that∇tη=ω.

To this end, let r ∈ N be an integer such that wJ = 0 for all J with J∩ {1, . . . , r−1} 6=∅(which is an empty condition forr= 1). We prove that we can find ap-formη with coefficients inA^?D_X ^t

t such that

(14) ω− ∇tη

∈ X

J∩{1,...,r}=∅

A^?D_X ^t

t,ϑdxJ. Successive application of this argument will prove the claim.

By assumption

(15) 0 =∇tω=: X

#K=p+2

X

k∈K

sgnK(k)(Qkw_Kr{k}) dxK.

o

Takingk < randr∈J and examining the summand of (15) corresponding to the setK:={s} ∪J we see that

(16) QswJ= 0 for all such s= 1, . . . , r−1.

Now, consider the system (ΣJ)of partial differential equations for the un- known functionu_J, whereJ is a fixed subsetJ ⊂ {r, . . . , d}of cardinalityp+ 1 withr∈J:

(ΣJ) :







QsuJ = 0 for alls= 1, . . . , r−1 Q_ru_J =w_J,

together with the integrability assumption (16). Systems of this type had been studied by Majima ([8]) before. See for example [16], Appendix A for a presentation from which follows that this system always has a solution in A^<D_X ^t

t,ϑif the coefficients of the system belong to the same space of functions. In the case of moderate growth coefficients/solutions, the assertion follows from [5], Theorem A.1 (which is formulated for dimension 2 only, but generalizes one-to-one to the case of arbitrary dimension).

In each case, we can always find a solution uJ ∈ A^?D_X ^t

t,ϑ for any such J ⊂ {r, . . . , d} of cardinalityp+ 1 andr∈J and if we let

η:= X

Jas above

uJdxJ,

we easily see that (14) is satisfied.

Claim 2. — The local pairing (9) is perfect in the derived sense (i.e. the induced morphism

DR^modD_X ^t

t (e^gt)→RHomX_t(DR^<D_X ^t

t (e^−gt),e!C) and the analogous one with DR^<D_X ^t

t and DR^modD_X ^t

t exchanged are isomor- phisms).

Proof. — According to Claim 1both complexes have cohomology in degree 0 only, i.e. we are reduced to look at the pairing of sheaves

S^modDt⊗^∨S^<Dt →e!CU_t

with S^modDt := H⁰(DR^modD_X ^t

t (e^gt)) and ^∨S^<Dt := H⁰(DR^<D_X ^t

t (e^−gt)).

Again, we consider the local situation Dt = {x1· · ·xk = 0} and gt(x) = x^−m₁ ¹· · ·x^−m_k ^ku(x), which we restrict to a small enough open polysector V ⊂Xt

→b X_t. We define the Stokes multi-directions ofg_t alongD_t inside V to be

Σ^D_gt^t :=St⁻¹ ÅÅπ

2,3π 2

ãã ,

where St:Dt∩V →R/2πZdenotes the map St(ri, ϑi) :=−

k

X

i=1

miϑi+ arg(u◦b(ri, ϑi)).

Let V_gt := (V rD_t)∪Σ^D_g^t

t , i.e. V_gt ∩D_t are the directions in which e^gt(x) has rapid decay for xradially approaching Dt. If eg_t : Vg_t ,→ V denotes the inclusion, one obviously has (possibly after shrinking V):

(17) S^modDt|_V =e_−gt!(e^−gt(x)·CUt)|_V and ^∨S^<Dt|_V =e_gt!(e^gt(x)·CUt)|_V. Consequently we have the following commutative diagram:

(18)

RHom(S^modDt,e!CU_t)|V −−−−→ ^∨S^<Dt|V

∼=



 y



 y

∼=

RHom (e−g_t)!CVrD_t,e!CV∩U_t

|V −−−−→ (eg_t)!CV_gt

By the factorizatione=e_−gt◦eι_−gt witheι_−gt :V ∩U_t,→V_−gt, we see that RHom (e_−gt)!CV_−gt,e!CU_t∼= (e_−gt)_∗RHom CV_−gt,(eι_−gt)!CV∩Ut

∼= (e_−gt)_∗Hom CV−gt,(eι_−gt)_!CV∩Ut

= (e_gt)_!CV_gt, since(V rVg_t)∩Dtcoincides with the closure ofV_−gt∩Dtinside Dt. Hence, the bottom line of (18) is an isomorphism and thus

∨S^<Dt ∼=RHomXt S^modDt,e!CUt

locally on X_tover an arbitrary point of D_t. Interchanging^∨S^<Dt andS^modDt gives the analogous isomorphism.

Claim 3. — There is a canonical isomorphism C_X^rd_t(e^−gt)∼= DR^<D_Xt ^t(e^−gt)[2d]

in the derived category D^b(CXt), whered= dim_C(X_t).

Proof. — Here, the arguments of [5], Theorem 3.6, apply directly, we therefore only briefly give the main line of the proof: By Claim1 we have

DR^<D_X ^t

t (e^−gt)←− H^{∼ 0}(DR^<D_X ^t

t (e^−gt)) =:^∨S^<Dt. Additionally, there is a canonical morphism

C⁻·

Xt,Dt⊗^∨S^<Dt −→ C_X^rd_t(e^−gt),

since for any open V ⊂ Xt and an asymptotically section σ ∈ ΓV(^∨S^<Dt), the sectionσ∈^∨S^<Dt(V)⊂e_∗(e^gtCU_t)is rapidly decaying along any chain in c∈ C⁻·

X_t,D_t(V), hencec⊗σ∈Γ(V,C_X^rd_t(e^−gt)). We prove that this morphism is a quasi-isomorphism. Restricted to U_t⊂X_t, this is obvious.

o

Next, consider a small open polysector V around some ϑ ∈ b⁻¹(x0) with x0∈Dt. Let Σ^D_±g^t_t ⊂b⁻¹(x0)denote the set of Stokes-directions ofe^±gt as in the proof of Claim1.

Ifϑ∈Σ^D_−g^t

t then forV being a small enough polysector, we have^∨S^<Dt|_V = e_! e^gtCUt

|_V. For a smooth topological chainc inX_t, the local sectione^gt will not have rapid decay alongcinV as required by the definition unless the chain does not meetDt∩V. Hence

C_X^rd

t(e^−gt)|_V =C⁻·

Xt,Dt⊗^∨S^<Dt|_V.

If ϑ∈ Σ^D_gt^t, we can assume thatV is an open polysector such that all the arguments of points in V are contained in Σ^D_gt^t. Then ^∨S^<Dt|V ∼=e∗(e^gtCV).

Similarly, all twisted chains c⊗e^gt will have rapid decay inside V and again both complexes considered are equal toC⁻·

X_t,D_t⊗e_∗(e^gtCU_t).

Finally, if ϑseparates the Stokes regions ofe^gt and e^−gt, we have with the notation of (17):

∨S^<Dt|V ∼= (egt)!(e^gtCUt)|V.

The subspace Vg_t is characterized by the property that Vg_t ∩Dt consists of those directions along which e^gt(x) has rapid decay for xapproaching Dt. In particular, c⊗e^gt is a rapid decay chain on V if and only if the topological chain cin X_t approachesD_t∩V inV_gt at most. Hence

C_X^rd_t(e^−gt)|V =C⁻·

X_t,D_t⊗(eg_t)!(e^gtCU_t) =C⁻·

X_t,D_t⊗^∨S^<Dt|V. In summary, we have the following composition of quasi-isomorphisms

∨S^<Dt[2d]−→^' CX_t[2d]⊗^∨S^<Dt −→ C^{' −}·

X_t,D_t⊗^∨S^<Dt −→ C^' _X^rd_t(e^−gt).

In order to prove Theorem 2.7, consider the resolution A^<D_X ^t

t ⊗_b−1O_Xtb⁻¹Ω^r_X

t ,→ P_X^<Dt

t ⊗_b−1C^∞

Xtb⁻¹Ω^∞,(r,·⁾

Xt , ∂ , where P_X^<Dt

t denotes the sheaf of C^∞-functions flat at b⁻¹(Dt) and Ω^∞,(r,s)_X

t

denotes the sheaf ofC^∞forms onXtof degree(r, s). We then have the bicom- plex

Rd·^,·:= P_X^<Dt

t ⊗_b−1C^∞

Xt b⁻¹Ω^∞,(·^,·⁾

X_t , ∂, ∂ , whose total complex Rd· computes is a fine resolution ofe!CU_t.

According to Claim 1, the local duality pairing (9) reads as ^∨S^<Dt ⊗ S^modDt →e!CU_t^an. We now have a canonical quasi-isomorphism

β:C⁻·

Xt,Dt⊗ Rd·−→^' Db^rd,−·

Xt

of complexes mapping an elementc⊗ρ∈ C_X^−r

t,Dt⊗ Rd^s(V)of the left hand side over some openV ⊂X_tto the distribution given byη7→R

cη∧ρfor a test form