Dwork families and D-modules

DWORK FAMILIES AND D-MODULES ALBERTO CASTA˜ NO DOM´ INGUEZ Abstract. A Dwork family is a one-parameter monomial deformation of a Fermat hypersurface. In this paper we compute algebraically the invariant part of its Gauss-Manin cohomology under the action of certain subgroup of automorphisms. To achieve that goal we use the algebraic theory of D-modules, especially one-dimensional hypergeometric ones. 1. Introduction Fix kto be an algebraically closed field of characteristic zero. Let nbe a positive integer and let w= (w0, . . . , wn)∈Zn+1 >0be an (n+1)-tuple of positive integers such that gcd(w0, . . . , wn) = 1. We will denote by dnthe sum Pwi≥n+1. Let us consider the family, parameterized by λ∈A1= Spec (k[λ]), of projective hypersurfaces of Pn= Proj (k[x0, . . . , xn]) given by: xdn 0+. . . +xdn n−λxw0 0·. . . ·xwn n= 0 ⊂Pn×A1. This family is an example of a Dwork family, consisting of the deformation of a Fermat hypersurface by an arbitrary monomial in every variable. We will denote it by Xn,w. By pnwe will mean the restriction of the second canonical projection Pn×A1→A1to Xn,w. Any other family of the form a0xdn 0+. . . +anxdn n−bλxw0 0·. . . ·xwn n= 0 ⊂Pn×A1, which a priori may appear more general, is actually isomorphic to one of the previous by the homography xi7→ ηixi,λ7→ bQη−wi iλ, where ηiis a dn-th root of ai. 1.1. A bit of history. Dwork families were introduced by Dwork, especially the case Xw,1, as a very interesting tool to understand the behaviour of the zeta function of a hypersurface defined over a finite field under a deformation (cf., for example, [Dwo63]). Until the nineties, the most important contributions to the knowledge of this families were due to Katz, in [Kat72,Kat80], where respectively, he studied the differential equation satisfied by the class of the form generating the middle Gauss-Manin cohomology and showed the very interesting relation between that cohomology in the `-adic context and the L-function of generalized Kloosterman sums. For more information about the origins of the work with Dwork families, see the historical introduction of [Kat09]. Since that discovery, very little was done with Dwork families until the introduction of mirror symmetry. Some of such families were a nice example of Calabi-Yau manifolds and turned out to be of interest to physicists working in that field (cf. [CORV00]). Because of this new interest on them, many author rediscovered Dwork families as a good tool to deal with other problems. We can cite, for instance, [DS04], studying the quantum cohomology of weighted n-dimensional projective spaces or [HSBT10], tackling the Sato-Tate conjecture in generality, among other results. The problem of calculating some part of the Gauss-Manin cohomologies of the Dwork family Xn,w or the whole of it has been also addressed. We can classify the diverse works dealing with such problem by its arithmetical setting and its way of attacking the problem. From the `-adic point of view, in [Kat09] all the cohomologies of Xn,w are computed by using all the power of the ´etale machinery. The same techniques are used in [RLW07], as a way to find the moment zeta functions for the original Dwork family and its quotient under the action of certain group of automorphisms. These works are quite complete and make the most of the `-adic approach. 2010 Mathematics Subject Classification. Primary 14D05, 14F10. Key words and phrases. D-modules, Gauss-Manin systems, Dwork families, hypergeometric D-modules. The author is partially supported by FQM218, P12-FQM-2696, MTM2013-46231-P, ERDF, ANR-13-IS01-0001-01/02 and DFG grants HE 2287/4-1 and SE 1114/5-1. 1 2 ALBERTO CASTA˜ NO DOM´ INGUEZ In the p-adic setting, a significative part of the work is carried out referring to Dwork’s classical methods of p-adic analysis. We can cite [Klo07], giving the expression of the matrix of the associated integrable Gauss-Manin connection using a mixture of rigid and Monsky-Washnitzer cohomology theories and Dwork’s techniques, or [Yu09], studying the variation of the unit root of the original Dwork family, also with methods going back to Dwork with a touch of crystalline cohomology. Over the complex numbers, we have a different approach, computational but still after Dwork and Katz, in [Sal13]. Also Yu and Katz treat this setting in their respective papers [Yu09] and [Kat09] above referred to. The former finds an horizontal section of the Gauss-Manin connection, whereas the latter expresses the same invariant part of the Gauss-Manin cohomologies issued in this paper in terms of hypergeometric D-modules using analytical transcendental methods. In this paper we deal with that last problem, but in a purely algebraic way by using the power of the theory of D-modules over any algebraically closed field of characteristic zero. This work can be seen as a first step in two directions. Namely, one is finding, as Katz and Kloosterman do in their papers but in an algebraic way as in the present paper, the expression of all the Gauss-Manin cohomologies of a Dwork family, not only some invariant part. The second is the extension of this work to the p-adic world, with the help of a good theory of p-adic cohomology (cf. [AC18] or [Meb13] to see the two existing approaches). Before ending this introduction, let us introduce a couple of conventions. An algebraic variety, or just variety, will mean for us a separated, finite type, equidimensional scheme over k, reducible or not. We will denote by πithe i-th canonical projection from a product, and by πXthe projection to a point from any variety X. 1.2. Some notions on D-modules. Before going on, we will digress a little and recall some notions from algebraic D-module theory that will be useful to state properly the main results of this paper (the references used here in this regard are [HTT08, Kat90, Meb89]). Associated with a smooth algebraic variety X, let us denote by OXand DXits structure sheaf and the sheaf of differential operators on it, respectively. We will denote by Db(DX) the derived category of bounded complexes of (left) DX- modules. We can also define the derived categories of bounded complexes of DX-modules with coherent, holonomic and regular holonomic cohomologies, each of them being a full triangulated subcategory of the precedent. Whenever we talk about a complex of DX-modules, we will understand it as an object of one of the derived categories above, which will be clear from the context. Associated with a morphism of smooth varieties f:X→Y, we will denote the usual direct and inverse image functors by f+and f+, and the extraordinary direct and inverse images by f!:= DYf+DX and f!:= DYf+DX, respectively, DZbeing the duality functor on a variety Z. Regarding tensor products, we will write ⊗L OXand for the interior and exterior ones, respectively. We will also use extensively the intermediate extension, a third image functor under some special conditions. For any open subvariety Uof Xconsider the inclusion j:U→X. If jis affine, the intermediate (or middle) extension of a holonomic D-module Mis the image of the canonical morphism j!M → j+M; it is denoted by j!+M(see more details in [HTT08, §3.4]). Recall that every holonomic D-module is both Noetherian and Artinian and then of finite length, so all of them admit a composition series. This allows us to define the semisimplification of a holonomic D-module as the direct sum of all of its composition factors. Another relevant functor in this work is Fourier (or Fourier-Laplace) transformation. It is a very important functor in D-module theory, of which we will only treat here its absolute one-dimensional version. It can be defined as an endofunctor of Db(DA1) given by FT = π2,+π+ 1• ⊗L OA2L, where L is the holonomic DA2-module DA2/(∂x−y, ∂y−x). Fourier-Laplace transformation preserves coherence over DA1and holonomy but not regularity; it is an equivalence of categories when defined on the associated full triangulated subcategories of Db(DA1) with the first two properties. We will say that a D-module is constant if it is formed as a successive extension of structure sheaves. Usually constant modules are not quite interesting, and it would be good to have some way of isolating the “nonconstant” part of a D-module. Here it is a way to do so in the concrete one-dimensional setting that is of interest to us. DWORK FAMILIES AND D-MODULES 3 Definition 1.1. Let Mbe a holonomic DA1-module, and let jbe the canonical inclusion of Gminto A1. We define its nonconstant part as NC M= FT−1j!+j+FT M. In the same vein, given a holonomic DGm-module M, we will define its nonconstant part as NC0M=j+NC j!+M. Remark 1.2.The functors NC and NC0are exact in the category of holonomic D-modules, respectively over A1and Gm, and preserve irreducible objects. An irreducible holonomic D-module has zero nonconstant part if and only if it is O. Therefore, for any holonomic D-module Mits nonconstant composition factors and those of NC Mare the same. Recall also that, if Xis of dimension nand Mis a complex of holonomic DX-modules, the Euler- Poincar´e characteristic of Mis χ(M) = (−1)nX k (−1)kdim HkπX,+M. Hypergeometric D-modules are a crucial ingredient of this work. Even though we will explain them in detail in section 2, let us just define them here. Definition 1.3. Let (n, m) be a couple of nonnegative integers, and let α1, . . . , αnand β1, . . . , βm, respectively, nand melements of kand γ∈k∗. The hypergeometric D-module associated with γ, the αiand the βjis defined as the quotient of DGmwith the ideal generated by the so-called hypergeometric operator γ n Y i=1 (D−αi)−λ m Y j=1 (D−βj). We will denote it by Hγ(α1, . . . , αn;β1, . . . , βm), or in an abridged way, Hγ(αi;βj). 1.3. Contents of the paper. Denote by (α1, . . . , αn) and (β1, . . . , βm) two unordered n- and mtuples (multisets), respectively, of elements from k. We will define their cancelation, denoted by cancel(α1, . . . , αn;β1, . . . , βm), as the result of eliminating from both tuples the elements that they share modulo Z, obtaining shorter disjoint lists (cf. [Kat90, 6.3.10]). In other words, assume that, up to a reordering on the αiand the βj, there exists an integer 0 ≤r≤min(n, m) such that αk≡βkmod Zfor k≤r, αk6≡ βkmod Zfor k > r. Then, under this assumption, cancel(α1, . . . , αn;β1, . . . , βm) = (αr+1, . . . , αn;βr+1, . . . , βm). If k= min(n, m) at least one of the tuples would be empty. Note that the subtractions remove only an element of the first tuple for each similar element of the second one, and thus the difference between the lengths of the resulting tuples is still the same. There exists an important subgroup of the group of automorphisms of Xn,w, being the one which will provide us our main object of study. Let µ0 dn=((ζ0, . . . , ζn)∈µn+1 dn n Y i=0 ζwi i= 1), and let G=µ0 dn/∆, the quotient of µ0 dnby the diagonal subgroup, acting linearly over Xn,w by componentwise multiplication on the projective variables. Let Xbe a quasi-projective variety and Uan open subvariety of the affine line. Given a smooth family of hypersurfaces X ⊂ X×U, together with the restriction of the second canonical projection p:X → U, we will define the Gauss-Manin complex of Xrelative to Uas the direct image p+OX, understood as a complex of regular holonomic DU-modules. The cohomologies of such complex, specially the middle one, are the Gauss-Manin cohomologies of X(relative to U). Our goal is to calculate the invariant part under the action of Gof the Gauss-Manin complex of Xn,w relative to the parameter λ, or in other words, pn,+OXn,w G. This must be understood as follows. 4 ALBERTO CASTA˜ NO DOM´ INGUEZ Gacts linearly over Xn,w, and in fact, over Pn×A1. Then for any G-equivariant DXn,w - or DPn×A1- module (cf. [Kas08, Def. 3.1.3], noting that quasi-G-equivariance and G-equivariance coincide due to the finiteness of G) we have an action of the Lie algebra associated with G, which, by the latter being finite and abelian, is the commutative group algebra k[G]. In our case, OXn,w is a G-equivariant DXn,w -module, and since pnis a G-equivariant morphism by definition, the direct image pn,+induces aG-equivariant structure on pn,+OXn,w (cf. [op. cit., p. 169]). In this sense, whenever we talk about the invariant part of a D-module M, we will understand its image by the functor Homk[G](k,•). In fact, if we focus on the nonconstant part of the cohomologies of pn,+OXn,w G, following Definition 1.1, we can state our first main result: Theorem 1.4. Let jbe the canonical inclusion from Gmto A1. Let Hnbe the irreducible hypergeometric D-module Hγncancel 1 w0 ,...,w0 w0 ,..., 1 wn ,...,wn wn ;1 dn ,...,dn dn. Then only the zeroth cohomology of pn,+OXn,w Ghas a nonconstant part, which is j!+ι+ nHn, with ιn:Gm→Gmbeing the map given by z7→ z−dn. As we said in the introduction, this result was obtained by Katz and Kloosterman in different contexts (cf. [Kat09, Thm. 5.3, Cor. 8.5] and [Klo07, Prop. 5.3], respectively). In that sense, Theorem 1.4 is the analogous statement in the algebraic case. However, we can go further and be more precise than them, as the following theorem shows. For each n, let us denote by Cnthe following set: Cn={k/dn:k/dn=j/wi, for some i= 0, . . . , n, j = 1, . . . , wi−1, k = 1, . . . , dn−1}. Note that this set will be empty if and only if dnis prime to each wi. Denote by ¯ Knthe Gauss-Manin complex of the smooth affine part of the quotient Xn,w/G (see Section 3 for more details). We will prove first Theorem 3.3 stating that pn,+OXn,w Gand ¯ Kndiffer only in some constant part, so we can focus on calculate ¯ Kn. We can give more information about it in our next main theorem, in such a way that Theorem 1.4 can be seen as a corollary of it: Theorem 1.5. The restriction j+¯ Knis the inverse image under ιnof another complex Kn∈Db rh(DGm), for which the following holds: HiKn= 0 if i /∈ {−(n−1),...,0},HiKn∼ =O(n i+n−1) Gmas long as −(n−1) ≤i≤ −1, and in degree zero we have the exact sequence 0−→ Gn−→ H0Kn−→ On Gm−→ 0, Gnlying in an exact sequence of the form 0−→ Hn−→ Gn−→ M α∈Cn Kα−→ 0, where Kαstands for the Kummer D-module DGm/(D−α). The statement of the theorem could seem a vague definition of a new complex Kn, but its point is giving in detail the expression of ¯ Knin a simple way. Nevertheless, we will define Knproperly in section 3. Note that [RLW07, Thm. 2.1] is the `-adic analogue of Theorem 1.5 for the original Dwork family, without mentioning the second exact sequence. In this sense our statement about the invariant part of the Gauss-Manin complex of the Dwork family is finer than any other result in any other context. The paper is structured as follows. Section 2 is a compilation of basic results on hypergeometric D-modules that will be of use in the following three sections. In section 3 we introduce properly our problem and start to tackle our main results. Next we exploit in section 4 the power of the inductive method used to prove most of Theorem 1.5, whereas in section 5 we finish its proof by finding the exponents of Gnat the origin and infinity. Acknowledgements. This work addresses a problem introduced by Steven Sperber to the author, to whom he is deeply grateful. The results of this paper conform an improvement of the main part of the doctoral thesis of the author, advised by Luis Narv´aez Macarro and Antonio Rojas Le´on. The DWORK FAMILIES AND D-MODULES 5 author wants to thank the latter for having the idea of the inductive framework, and both of them for their support and encouragement. He is also grateful to Claude Sabbah for suggesting him the use of Fourier transformation and pointing him out the reference [DS04], and an anonymous referee for the useful comments given to improve the paper. He wants to thank as well his son, Alberto Luna Casta˜no, born during the early writing of this paper, for making him learning many new things and improving his productivity. This work is dedicated to him. 2. Hypergeometric D-modules In this section we introduce hypergeometric D-modules, studying their basic properties and their parameters, ending by characterizing such a D-module by them. Almost any result stated here is inspired by or directly equal to some other of [Kat90, §3]. In fact, apart from Katz’s reference, we have not found in the literature an algebraic approach to hypergeometric D-modules as his. Throughout the text, we will remain with one-dimensional D-modules. From now on, we will denote by Dλthe product λ∂λ, omitting the variable as long as it is clear from the context. Proposition 2.1. Recall that a Kummer D-module is the quotient Kα=DGm/(D−α), for any α∈k. Let Mbe a holonomic DGm-module. Then its Euler-Poincar´e characteristic is zero if and only if its composition factors are Kummer D-modules. Proof. The Euler-Poincar´e characteristic of a holonomic DGm-module is additive and never positive, so we just need to show the equivalence when Mis irreducible. If M=Kα, since it has no singularities at Gmwe can apply Deligne’s formula at [Kat90, 2.9.8.2]. In this case, Mhas a regular singularity both at zero and infinity, so χ(M) = 0. Suppose then that χ(M) = 0. By [op. cit., Cor. 2.9.6.1], we have that M, being irreducible, coincides with j!+j+M, for any open subvariety Uj ,→Gmof Gmin which Mis a module with integrable connection. Then we can apply now Deligne and Gabber’s formula [op. cit., Thm. 2.9.9]. Since χ(OGm) = 0, we have necessarily that all the singularities of Mare regular, and, following Katz’s notation, droppM= 0 at every point of Gm−U. Therefore Mcan be seen as a module with integrable connection on Gm, so there exists a square matrix Aof elements of k[λ±] of order rsuch that M∼ =Dr Gm/(D−A). Now we can apply to this connection the theorem of the cyclic vector [DGS94, Thm. III.4.2] to obtain that Mis isomorphic to DGm/(P(λ, D)), with Pbeing a nonconstant polynomial of k[y±, t]. (Note that although the proof of the theorem of the cyclic vector requires the connection to be defined over a function field of the form k(λ), it only actually needs that λis invertible, as in our case, providing a global proof in Gm.) Now Mhas a regular singularity at zero, so the degree in λof the coefficients of Pcannot be negative (cf. [Del70, 1.1.2]). Taking µ=λ−1, we have Dµ=−Dλ, so by the same argument, the order at µof the coefficients of Pmust be zero at least too, and thus Pis a polynomial only in D. Since kis algebraically closed and Mis irreducible, deg P(D) = 1, for if it were greater we would have a composition factor of Mconsisting of the quotient of DGmby the left ideal generated by any factor of P(D). In conclusion, M∼ =DGm/(D−α).  We have just characterized, up to semisimplification, the holonomic DGm-modules of Euler-Poincar´e characteristic zero. Regarding characteristic −1, we can claim that the composition factors of any holonomic DGm-module of Euler-Poincar´e characteristic −1 will be a finite amount of Kummer D- modules and an irreducible DGm-module of characteristic −1, which can be an irreducible punctual DGm-module supported at any point of Gm, also called delta D-module. Irreducible hypergeometric D-modules of positive rank, as we will see, are the irreducible nonpunctual holonomic DGm-modules of characteristic −1. Let us return to them. Remark 2.2.The special type (n, m) = (0,0) in Definition 1.3 corresponds to delta D-modules on Gm, since Hγ(∅;∅) = DGm/(γ−λ). On the other hand, is easy to check that Hγ(αi;βj)⊗OGmKη∼ =Hγ(αi+η;βj+η). In this paper we will only focus on the case n=m, so that Hγ(αi;βj) has only regular singularities, at the origin, infinity and γ, where the Jordan decomposition of its local monodromy (when k=C) is a pseudoreflection, that 6 ALBERTO CASTA˜ NO DOM´ INGUEZ is, the space of formal meromorphic solutions Hom(H,k((xγ))) is (n−1)-dimensional (cf. [Kat90, Lem. 2.9.7]. Any hypergeometric D-module is of Euler-Poincar´e characteristic −1 ([Kat90, Lem. 2.9.13]) and its behaviour at P1can be fully understood from its parameters, as we are going to show in the following results. Proposition 2.3 (Irreducibility).(cf. [Kat90, Props. 2.11.9, 3.2]) Let H:= Hγ(αi;βj)be a hypergeometric D-module. It is irreducible if and only if αi−βjis not an integer for any pair (i, j)of indexes. The next proposition deals with the exponents of hypergeometric D-modules. As we will shortly see, they are an important property in order to characterize them. For more information on the exponents of a D-module in dimension one, see [CD17, §2]. Proposition 2.4 (Exponents).(cf. [Kat90, Cor. 3.2.2]) Let H:= Hγ(αi;βj)be an irreducible hypergeometric D-module of type (n, m), and fix a fundamental domain Iof k/Z. Then, i) The Jordan decomposition of the regular part of Hat the origin is (H ⊗ k((λ)))slope=0 ∼ =M α∈I k((λ))[Dλ]/(Dλ−α)nα, where nαis the amount of αicongruent to αmodulo Z. ii) The Jordan decomposition of the regular part of Hat infinity is (H ⊗ k((µ)))slope=0 ∼ =M β∈I k((µ))[Dµ]/(Dµ−β)nβ, where µ= 1/λ and nβis the number of βicongruent to βmodulo Z. Proposition 2.5 (Isomorphism class).([Kat90, Lem. 3.3]) Let H=Hγ(αi;βj)be a hypergeometric D-module of type (n, m). Its isomorphism class as DGm-module determines nand m, the set of all of the αiand βjmodulo Z, and if either His irreducible or n=m, the point γ. As a matter of fact, this proposition shows that in the regular or the irreducible case, all the parameters of a hypergeometric D-module are intrinsic to it. Namely, assuming n≥m,nis the generic rank of H,mthe rank of its slope zero part at infinity, the αiand the βjare the exponents at the origin and infinity, respectively, and γis the other regular singularity of Hin the case it is regular. The argument in the irreducible irregular case is a bit more complicated, but we will not need it in the following. Proposition 2.6. (cf. [Kat90, Thm. 3.7.1], [LS91, Thm. 1]) Let Mbe a nonpunctual irreducible holonomic DGm-module of Euler-Poincar´e characteristic −1. Then Mis a hypergeometric D-module. 3. Setting the problem Once we have introduced the objects to study and the tools to work with, let us keep on and start to work on the problem addressed in the paper. Proposition 3.1. Xn,w is a smooth quasi-projective variety. Write γn= (dn)−dn n Y i=0 wwi i. Then the morphism pnis smooth over the open subvariety Un=λ∈A1|γnλdn6= 1. Proof. Let us use the multi-index notation, for the sake of simplicity, in such a way that xawill represent the monomial xa0 0·. . . ·xan nfor any (n+ 1)-uple aof positive integers. The partial derivatives of xdn 0+. . . +xdn n−λxwwith respect to the xiand λare δi:= dnxdn−1 i−λwixw−eiand δλ:= −xw, respectively. If δλ= 0, then some ximust vanish, and if δi= 0 for every i, all of the xiwill be zero, which is impossible. Therefore, Xn,w is smooth. Since at the singular points of the fibers of pn DWORK FAMILIES AND D-MODULES 7 we have that xi6= 0 for every i, we can multiply by them the partial derivatives δi, obtaining that dnxdn i=−wiλxwfor every i, so w0xdn i=wixdn 0. But then, substituting the xdn iin the equation of Xn,w, (dn/w0)xdn 0=λxw. Taking dn-th powers at each side, we get that ddn n=λdnQiwwi i, so if we do not have that equality, we will find ourselves at a nonsingular fiber.  Recall that we had an action of the group Gon the Dwork family Xn,w. Since Gis a finite group, Xn,w/G is another quasi-projective variety, isomorphic to x0+. . . +xn=λxn+1 xw0 0·. . . ·xwn n=xdn n+1 ⊂Pn+1 ×A1. Substituting x0in the second equation for its value given by the first one and taking xn+1 = 1, we find that Xn,w/G is the projective closure of Yn,w :xw1 1·. . . ·xwn n(λ−x1−. . . −xn)w0= 1 ⊂Gn m×A1= Spec kx± 1, . . . , x± n, λ. In this sense we will write ¯ Yn,w for Xn,w/G. Let Zn,w ⊂An×A1= Spec (k[x1, . . . , xn, λ]) be the variety defined by the equation xw1 1·. . . ·xwn n·(1 −x1−. . . −xn)w0=λ. We will omit the (n+ 1)-tuple wfrom Zn,w when it is clear from the context. Denote by 0pnand 00pnthe projections from Yn,w and Zn,w, respectively, to A1. Then we can form the following cartesian diagram: (1) Yn,w −0p−1 n(0)  0pn  ˜αn//Zn,w −00p−1 n(0) 00pn  Gm ιn//Gm , where ˜αn(x, λ) = (x1/λ, . . . , xn/λ), λ−dn(note that it is the restriction of an endomorphism of Gn+1 m). Proposition 3.2. The families Yn,w and Zn,w are smooth, but not their projective closures. The projections 0pnand 00pnare smooth, respectively, over the open subvarieties Unand Gm− {γn}. Proof. Yn,w and Zn,w are smooth for being, respectively, a n-dimensional torus and a graph. However, their projective closures will have singularities at their sections at infinity independently of λ, because both of them are the cartesian product of the same arrangement of hyperplanes with A1. Regarding the fibers of 0pn, since the section at infinity of ¯ Yn,w is independent of λ, the singular fibers of Yn,w will be over the same points of A1as those of ¯ Yn,w. Now the quotient map πG:Xn,w →¯ Yn,w is G-equivariant by definition, as well as pn. Then the singular locus of 0pn:¯ Yn,w →A1is the same as that of pn:Xn,w →A1, which is Un. With respect to the fibers of Zn,w, note that ˜αnis an ´etale morphism out of its section with equation λ= 0, where it is ramified. Therefore, Zn,w will have nonsingular fibers on the image by ιnof Un except the origin, that is to say, Gm− {γn}. Recall that we wanted to compute the invariant part under the action of Gof the Gauss-Manin complex of Xn,w relative to the parameter λ, that is, pn,+OXn,w G. Remember as well that what is actually more interesting to us is the nonconstant part of its cohomologies. We can actually restrict ourselves to an affine context in order to find it: Theorem 3.3. Let ¯ Kn=0pn,+OYn,w . There exists a canonical morphism between the complexes of DA1-modules pn,+OXn,w G−→ ¯ Knsuch that the cohomologies of its cone are direct sums of copies of the structure sheaf OA1. Proof. Since Xn,w is smooth, RΓ[Xn,w]OPn×A1∼ =ι+OXn,w [−1], where ιdenotes the inclusion Xn,w ,→ Pn×A1(cf. [HTT08, Prop. 1.7.1]). The action of Gon Xn,w can be easily extended to Pn×A1, in such a way that ιis a G-equivariant morphism. Then, as discussed in subsection 1.3, we can claim by [Kas08, p. 169] that RΓ[Xn,w]OPn×A1is a G-equivariant DPn×A1-module. 8 ALBERTO CASTA˜ NO DOM´ INGUEZ Let us see now ¯ Yn,w as a quasi-projective variety in Pn×A1and call M:= RΓ[¯ Yn,w]OPn×A1. Let JXand JYbe the ideals of definition of Xn,w and Xn,w/G, respectively. The invariant part under the action of Gof the rings OPn×A1/Jk Xare, by construction, OPn×A1/Jk Y, for every k≥0 (cf. [Har92, p. 127]). Since we are working with a finite abelian group and sheaves of k-vector spaces, we can claim thanks to Maschke’s theorem that the functor •G= Homk[G](k,•) is exact. Furthermore, Gis finite, and thus isomorphic to the product of some cyclic groups. Then the invariant part of a sheaf of k-vector spaces is the kernel of the product of the linear maps ϕai−id, the aiand ϕaibeing the generators of Gand their associated actions on the sheaf. Since •Gis a kernel and an exact functor, it commutes with derived functors of left exact ones. In particular, so it does with RHomOPn×A1(•,OPn×A1), and then the invariant part of RΓ[Xn,w]OPn×A1under the action of Gmust be M. By the same arguments as in the beginning of the proof, π2,+RΓ[Xn,w]OPn×A1[1] ∼ =pn,+OXn,w . Let us prove now that taking invariants and direct image by π2commute, so that pn,+OXn,w G=π2,+M[1]. The morphism π2is a projection, so the functor π2,+is the image by Rπ2,∗of the relative de Rham complex DRπ2shifted ndegrees to the left. By the same reasons as in the previous paragraph, Rπ2,∗ commutes with •G. The relative de Rham complex is a complex of sheaves of k-vector spaces whose objects are N ⊗OPn×A1Ωi Pn×A1/A1for some DPn×A1-module N. The connecting morphisms are k-linear, and then G-equivariant. Since locally the differential modules are isomorphic to a direct sum of copies of OPn×A1, the objects of the relative de Rham complex will be locally isomorphic to a direct sum of copies of N. Consequently, •Gand DRπ2will commute as well and then, pn,+OXn,w G=π2,+M[1]. Let now Y∞ n,w be the intersection of the hyperplane at infinity with ¯ Yn,w and denote by i:Pn−1×A1→ Pn×A1and j:An×A1→Pn×A1the canonical immersions. We have the associated excision distinguished triangle π2,+M −→ π2,+j+M −→ π2,+i+M −→ . Now, by [Meb89, Prop. I.6.3.1], i+M∼ =RΓ[Y∞ n,w]OPn−1×A1∼ =π+ 1RΓ[¯ A]OPn−1, ¯ Abeing the projective arrangement of hyperplanes such that Y∞ n,w is the product ¯ A×A1. Then π2,+i+M is the tensor product πPn−1,+RΓ[¯ A]OPn−1⊗kOA1and it has constant cohomologies. Finally, j+M∼ =RΓ[Yn,w]OAn×A1. Since Yn,w is smooth, we have π2,+j+M∼ =¯ Kn[−1], so in the end we have a triangle pn,+OXn,w G−→ ¯ Kn−→ π2,+i+i+M[1] −→, the last complex having direct sums of copies of the structure sheaf as cohomologies, and we are done.  Remark 3.4.The morphism pnis proper and smooth and OXn,w can be endowed with the structure of a pure Hodge module. Then, by virtue of [Sai90, 4.5.2-4.5.4], we can affirm that pn,+OXn,w can be upgraded to a semisimple complex of Hodge modules, being in such a case the direct sum of its cohomologies, which are in turn semisimple Hodge modules. The same applies to their underlying DA1-modules. Now note that pn,+OXn,w Gis a direct summand of pn,+OXn,w , because the latter can be decomposed as the direct sum of its eigenspaces associated with the action of G, so in particular, the former is a semisimple complex of DA1-modules too. In particular, the theorem tells us that the nonconstant part of the cohomologies of pn,+OXn,w G are those of ¯ Kn, which coincide with the middle extension of their restrictions to Gm. Note indeed that due to the theorem, ¯ Knhas the same singularities at the origin as pn,+OXn,w G, which as said in the paragraph above, is a direct summand of pn,+OXn,w which has in turn no singularity at the origin, for pnis smooth there. Summing up, the nonconstant part of the cohomologies of ¯ Kncannot have any punctual composition factor at the origin. Let us continue now with the proof of Theorem 1.4, by using the theorem above. We need some more information about the Fourier transform of ¯ Knand we will gather it from the work [DS04] by Douai and Sabbah. DWORK FAMILIES AND D-MODULES 9 Proposition 3.5. (cf. [DS04, Prop. 3.2]) Let n≥1. Then we have an isomorphism of complexes of DGm-modules j+FT ¯ Kn∼ =[dn]+Hddn nγn1 w0 ,...,w0 w0 ,...,wn wn ;∅. Definition 3.6. Let M ∈ Db c(DA1) be a complex of coherent DA1-modules, and let jbe the canonical inclusion of Gminto A1. We define its Fourier localization as FLoc M= FT−1j+j+FT M. Remark 3.7.The functor FLoc is exact in the category of coherent DA1-modules. In addition, for any holonomic DA1-module M, there exist two canonical morphisms M −→ FLoc Mand NC M −→ FLoc Mwhose kernels and cokernels are constant (we have the first one as well if Mis only coherent). Proof of Theorem 1.4. Since j+FT ¯ Knis a single DGm-module at degree zero, by the previous remark all the other cohomologies of ¯ Kn, and then of pn,+OXn,w G, must be constant, so we can focus on the zeroth one. After Remark 3.4 and Proposition 3.5, we just need to show that ι+ nHnis isomorphic to NC0j+H0¯ Kn, which is isomorphic to j+NC H0¯ Knbecause of H0¯ Knnot having singularities at the origin. We know that j+FT ¯ Kn∼ =DGm/(P), where Pis γn n Y i=0 wi Y j=1 D−dnj wi−λdn. Therefore, applying FT−1to the canonical morphism DA1/(P)→j+j+FT ¯ Kn, we can deduce that there exists another canonical morphism from M:= FT−1DA1/(P) = DA1/(Q) to FLoc ¯ Kn= FLoc H0¯ Kn. We can choose Qas minus the image by FT−1of P, that is, Q=∂dn−γn n Y i=0 wi Y j=1 ∂λ +dnj wi. Since the kernel and the cokernel of the morphism DA1/(P)→j+j+FT ¯ Knare necessarily supported at the origin, then both those of M → FLoc ¯ Knmust be constant. In other words, we can form an exact sequence of the form 0−→ Or A1−→ M −→ FLoc H0¯ Kn−→ Os A1−→ 0, for some r, s ≥0. Since NC FLoc = NC, applying now j+NC to that sequence we obtain the isomorphism j+NC M∼ =j+NC H0¯ Kn. Now note that Mhas no singularities at the origin, so j+NC M∼ =NC0j+Mas well. Then we just need to compute j+M, but then we can multiply Qon the left by λdn, and since λdn∂dn=Qdn−1 j=0 (D−j), we have in conclusion that j+NC H0¯ Knis isomorphic to the nonconstant part of the quotient of DGm by the left ideal generated by dn−1 Y j=0 (D−j)−γnλdn n Y i=0 wi Y j=1 ∂λ +dnj wi, which is isomorphic to the inverse image by ιnof the hypergeometric DGm-module Hγn1 w0 ,...,w0 w0 ,..., 1 wn ,...,wn wn ;1 dn ,...,dn dn. This hypergeometric D-module has Hnamong its composition factors, together with every Kummer D-module of the form Kαwith α∈Cn(cf. [Kat90, Cor. 3.7.5.2]). However, all those Kummer factors are sent to OGmby ι+ n, so the proof is finished.  Define the complex Knto be pn,+OZn−p−1 n(0) ∈Db rh(DGm). It fits with the statement of Theorem 1.5 by applying base change using diagram 1 (cf. [HTT08, Thm. 1.7.3]). To prove that theorem we should compute then the Gauss-Manin complex of the family Znoutside the origin. Thanks to a combination mainly of Proposition 2.5 and Proposition 2.6 we can characterize Gnup to semisimplification if we prove that its Euler-Poincar´e characteristic is -1, find its generic rank as OGm-module, calculate the 16 ALBERTO CASTA˜ NO DOM´ INGUEZ to it. It is easy to check that independently on a6=dn−1and the position of the hypergeometric and Kummer composition factors on the exact sequence of Gn−2, H0πGm,+Gn−2⊗ K−a/dn−1∼ =H−1πGm,+Gn−2⊗ K−a/dn−1⊕k, so that we can find at H0Mna copy of each Ka/dn−1more than at H−1Mn. In order to determine which Kummer D-modules appear at H0Mn(and then at H−1Mntoo), we will apply [CD17, Thm. 1.1] and try to study the surjectivity of the map Φ = (f−t, ∂1+f0 1ϕα, . . . , ∂n−1+ f0 n−1ϕα) : Rn→Rfor non-integer values of α, where R=k((t))[x1, . . . , xn−1], f=xw1 1·. . .·xwn−1 n−1(x1+ . . . +xn−1)w0and ϕα=∂t+αt−1. To see that we can assume, without loss of generality, that we take c∈R, homogeneous in the xiof degree m≥0, and a, b1, . . . , bn−1∈R, homogeneous too of degrees m, m + 1, . . . , m + 1, respectively, such that Φ(a, b1, . . . , bn−1) = c. Then what we are assuming reads            fa + n−1 X i=1 f0 iϕαbi= 0 −ta + n−1 X i=1 ∂ibi=c . Thanks to Lemma 4.2, from the first equation we know that there exist homogeneous polynomials F, g(i,j)∈Rfor every 1 ≤i<j≤n, of respective degrees mand m−1 in the xi(so that the g(i,j) must be zero if m= 0), such that (7) a=−dn−1F ϕαbi=xiF+X j6=i ε(j−i)xixjσ ff0 jg(i,j), i = 1, . . . , n, εbeing the sign operator. Note that ϕαis always bijective for the values of αunder consideration. Substituting those values in the second equation of 7 and applying the Euler relation for F, we get that (8) dn−1t+m+n dn−1 ϕ−1 αF+X 1≤i<j≤n L(i,j)g(i,j)=c, where L(i,j)= ((wj−wi)σ+ (w0−wi)xj−(w0−wj)xi+xilj∂i−xjli∂j)ϕ−1 α for each pair (i, j); they depend on αonly because they apply ϕ−1 α(which is always an isomorphism) to the g(i,j). Call Athe operator acting on F. Now if dn−1αis not an integer, Ais invertible, so the system has a solution. And when dn−1αis an integer the lack of surjectivity is independent of the choice of α, as we wanted to know; should equation 8 above have a solution, it would happen for any α, changing appropriately m. In conclusion, all the classes modulo Zof the elements of {1, . . . , dn−1−1}appear as exponents of H0Mnwith the same multiplicity. Then we can affirm that there exists some nonnegative integer rsuch that, for i=−1,0, HiMn∼ =O(n−1 i+n−2) Gm⊕ dn−1−1 M a=1 Kr+1+i a/dn−1. If we show that r= 0 we will have finished the proof. Recall formula 5 and how the Kummer summands are added to the last two cohomologies of Mn. At degree -1 they appear when tensoring some Kβ, for β∈Cn−2, with some Kα, for dn−1α∈Z. We have just seen by the calculation of the exponents that if r > 0 every Kummer Ka/dn−1should appear at H−1Mn. However, note that 1/dn−1can never be in Cn−2, for wi< dn−1for any i= 0, . . . , n −2. Then r= 0 and we are done.  In the proof above we have stated in 6 the two possibilities for the extension of Hn−2and the Kummer D-modules to obtain Gn−2.A priori it was not possible to determine its exact expression, but we can revisit the proof with all the information we know now about Mnand be much more precise: DWORK FAMILIES AND D-MODULES 17 Corollary 4.4. For each n > 2, we have an exact sequence of the form 0−→ Hn−2−→ Gn−2−→ M α∈Cn−2 Kα−→ 0. Proof. Recall the exact sequences of 6 for the index n−2: 0−→ Hn−2⊕M β∈C0 n−2 Kβ−→ Gn−2−→ M β∈C00 n−2 Kβ−→ 0, 0−→ M β∈C0 n−2 Kβ−→ Gn−2−→ Hn−2⊕M β∈C00 n−2 Kβ−→ 0. Take wn−1=dn−2, so that every β∈Cn−2appears as well as some a/dn−1for certain values of a∈ {1, . . . , dn−1−1}. Then, if C0 n−2were not empty, all of the H−1πGm,+(Kβ⊗ K−β)⊗ Kβ∼ =Kβ, for β∈C0 n−2, would be subobjects of H−1πGm,+(Gn−2⊗ K−β)⊗ Kβ, just by looking at the long exact sequence of cohomology of the triangle appearing from applying πGm,+(•⊗K−β)⊗ Kβto one of the exact sequences of 6. However, H−1πGm,+(Gn−2⊗ K−β)⊗ Kβ= 0, so necessarily C0 n−2is empty and C00 n−2is the whole Cn−2. Now we have two possibilities: either the extension of the Kummer Kβand the hypergeometric Hn−2 is trivial or it is not, corresponding respectively to the second and the first exact sequence of 6. Anyhow, we can write the exact sequence with Hn−2on the left-hand side, independently of whether it splits or not.  Even though the corollary provides us a piece of Theorem 1.5, we will not be able to use it in the following, since it lets us find the exact sequence of Gn−2, two steps behind the induction procedure. We will only use it at the very end of the actual proof of the theorem in section 5. We can also state an interesting small byproduct of Proposition 4.3, relating hyperplane arrangements: Corollary 4.5. For any n≥2, let A={H1, . . . , Hn+1}be a generic arrangement of n+1 hyperplanes in An(i. e., a central arrangement such that the intersection of every subset of nhyperplanes is the origin) with multiplicities w1, . . . , wn+1, sharing no common factor and denoting by dtheir sum. Then, the global de Rham cohomology of its Milnor fiber Fis HiπF,+OF∼ =k(n i+n−1),for −(n−1) ≤i≤ −1,and H0πF,+OF∼ =kn+d−1 Proof. Under a suitable affine change of variables, we can assume that the form defining our arrangement is f=xw1 1·. . .·xwn n(x1+. . .+xn)wn+1 . Then, since it is homogeneous, the Milnor fiber of the arrangement is defined by the equation f= 1. Let us denote by Tthe subvariety of An×Gmgiven by f−λ= 0, and consider the following cartesian diagram: F×Gm  π2  α//T π2  Gm[d] //Gm , where αis the morphism given by (x, λ)7→ (x1/λ, . . . , xn−1/λ, λd) and the morphisms denoted by π2 are actually the restriction to F×Gmor Tof the second projection of An×Gm. All the varieties involved are smooth, so by base change, [d]+π2,+OT∼ =π2,+OF×Gm∼ =πF,+OF⊗kOGm. Then the formula for the global de Rham cohomology of Ffollows just from the statement of the proposition, noting that Mn+1 ∼ =π2,+OT. Lemma 4.6. Let n≥2, and let αbe an element of ksuch that dn−1αis an integer. Then, π2,+(π2φn)+Kα∼ =OGm[1] ⊕ O2 Gmif α∈Z Kαotherwise. 18 ALBERTO CASTA˜ NO DOM´ INGUEZ Proof. For any α, a simple calculation shows that (π2φn)+Kα∼ =DGm×(Gm−{1})/(Dλ−α, ∂z−(dnz−wn)(z(1 −z))−1Dλ). Note that the ideal can be rewritten as (Dλ−α, ∂z−α(dnz−wn)(z(1 −z))−1), so that our D-module is in fact Kαk+H1(−αwn;−αdn), where k:Gm− {1},→Gmis the canonical inclusion. But then, π2,+(π2φn)+Kα∼ =Kα⊗πGm−{1},+k+H1(−αwn;−αdn). Note that (dn−wn)α=dn−1α∈Z. Then the question is now how to calculate the global de Rham cohomology of some k+H1(α;β), for α, β ∈ksuch that α≡βmod Z. However, due to the condition on αand β,k+H1(α;β)∼ =k+Kα. If α∈Z, then k+Kαis just OGm−{1}, so πGm−{1},+k+Kα∼ =k[1] ⊕k2. In order to treat the case α /∈Z, we will make use of [Kat90, Prop. 2.9.8] and extend our D-modules to Gm, for πGm−{1},+k+Kα∼ = πGm,+k+k+Kα. In that case, Kαis an irreducible DGm-module and has an integer exponent at 1 because it is regular there, so we can apply the aforementioned result to obtain the exact sequence 0−→ Kα−→ k+k+Kα−→ δ1−→ 0. Now since πGm,+Kα= 0, we get from the sequence that πGm,+k+k+Kα∼ =k. Proposition 4.7. For each n≥1,Knis concentrated in degrees −(n−1),...,0,HiKn∼ =Len a=1 K(n i+n−1) a/en for all −(n−1) ≤i≤ −1and in degree zero, we have an exact sequence of the form 0−→ Gn−→ H0Kn−→ en M a=1 Kn a/en−→ 0, where Gnis a DGm-module of Euler-Poincar´e characteristic −1, without any constant composition factor. Proof. Acting in the same way as in the proof of Proposition 4.3, let us assume that en= 1; the general case can be easily deduced from this particular one. The statement is an improvement of that of Proposition 3.9. To prove the part regarding the zeroth cohomology we only need to show that H0Kn has only ncopies of OGmas composition factors, appearing then all of them as a quotient, defining the Gnas the D-module with which we take such a quotient. To deal in a better way with the cohomologies of Kn, let us separate the copies of OGmfrom the semisimplification of the rest of composition factors, such that HiKnss =Oci⊕Gi, that is to say, we will denote by Githe direct sum of the nonconstant composition factors of HiKn. It is straightforward to deduce that HiKn= 0 for every i /∈ {−n, . . . , 0}decomposing λnas the graph immersion followed by the second projection. Following the spirit of the section, let us proceed inductively in nto find the ciand the Gi. If n= 1 we have already proved everything at Lemma 3.10 and Proposition 3.9, so let us go for the general case assuming that we know Kn−1. Remember that we have the distinguished triangle 3 Kn−→ π2,+(π2φn)+Kn−1−→ Mn−→ . Let us calculate its long exact sequence of cohomology. Using that (π2φ)+is an exact functor of D-modules and Lemma 4.1, we get an exact sequence for any iof the form (9) 0 → H0π2,+(π2φn)+HiKn−1→ Hiπ2,+(π2φn)+Kn−1→ H−1π2,+(π2φn)+Hi+1Kn−1→0. If any of the cohomologies of HiKnis formed, among others, by an extension of Oa Gmby Ob Gmfor certain aand b, it will necessarily be trivial (although that does not happen in general; consider for instance the extension 0 → OGm→ DGm/(D2)→ OGm→0). Indeed, by Theorem 3.3, the cohomologies Hij+¯ Kndo not have any singularity at the origin. Now take into account that those cohomologies are nothing but the image by ι+ nof those of Kn, so any of the extensions of Oa Gmthat we could have may be extended to the analogous over an affine line and then must be trivial, for Ext1 DA1(OA1,OA1) = 0. Take i∈ {−(n−2),...,−1}. We know that HiKn−1∼ =Len−1 a=1 K(n−1 i+n−2) a/en−1. By Lemma 4.6 above, (10) π2,+(π2φn)+ en−1 M a=1 Ka/en−1∼ =OGm[1] ⊕ OGm⊕ en−1 M a=1 Ka/en−1. DWORK FAMILIES AND D-MODULES 19 Therefore, we can claim that the exact sequences of the form 9 will split for any i6= 0,−1. Using that splitting and formulas 10, we can be more concrete when writing the long exact sequence of triangle 3, namely (11) 0→ H−nKn→0→0→ H−(n−1)Kn→ OGm→0→ . . . → HiKn→ O(n i+n−1) Gm⊕ en−1 M a=1 K(n−1 i+n−2) a/en−1→ en−1 M a=1 K(n−1 i+n−2) a/en−1→ . . . → H−1Kn→ H−1π2,+(π2φn)+Kn−1→ en−1 M a=1 K(n−1 n−3) a/en−1→ → H0Kn→ H0π2,+(π2φn)+Kn−1→ en−1 M a=1 Kn−2 a/en−1⊕ dn−1 M a=1 Ka/dn−1→0. Then H−nKn= 0, H−(n−1)Kn∼ =OGmand the Giwill be a sum of Kummer D-modules for i≤ −2. Denote by hj ithe dimension of the j-th global de Rham cohomology of Gi; since h−1 i= dim Γ Gm,RHomDGm(OGm, Gi), all of the h−1 imust vanish by definition of the Gi, and the h0 iwill vanish too for i≤ −2 because those Giare sums of Kummer D-modules. Notice now that the global de Rham cohomologies of Knand OZn are the same. The latter is already known thanks to [CD15, Prop. 5.2] because Znis the complement of an arrangement of hyperplanes in general position, so we will have the following system of equations:    ci+ci+1 =n+1 i+n, i =−n, . . . , −2 h0 −1+c−1+c0=n+1 n−1 h0 0+c0=n+ 1 , From these equations we get that for i=−(n−2),...,−1, c−i=n i+n−1, so that our system can be reduced to: h0 −1+c0=n h0 0+c0=n+ 1 . Nevertheless, the first point of Proposition 3.9 tells us that c0≥n, so the equality holds indeed and then h0 −1= 0 and h0 0= 1. Following an analogous argument, from the calculation of the ciwe deduce that rk HiKn≥n i+n−1at every degree; applying now the second point of Proposition 3.9 the equality holds again for i≤ −2 and so Gi= 0 for those values of i. Then we can also claim that every row of the long exact sequence 11 but the last two is a single short exact sequence in itself, all of them with the zero module at the beginning and the end. This proves everything (so that Gss n=G0) except the vanishing of G−1. We know that its rank is one at most, thanks to point ii of Proposition 3.9. Then it will be a Kummer D-module, for h−1 −1=h0 −1= 0. Let us focus first in knowing better H−1π2,+(π2φn)+Gn−1. From the exact sequence 0−→ Gn−1−→ H0Kn−1−→ en−1 M a=1 Kn−1 a/en−1−→ 0 and diagram 9 we can deduce that its nonconstant composition factors, together with some Kummer D-modules, are composition factors of H−1π2,+(π2φn)+Kn−1, which finds itself in an exact sequence between H−1Knand a sum of Kummer modules. Therefore, H−1π2,+(π2φn)+Gn−1is also an extension of some Kummer D-modules. By applying the functor π2,+(π2φn)+to any of the exact sequences from 6, we can claim that it seats in an exact sequence between H−1π2,+(π2φn)+Hn−1⊕ Lα∈C0 n−1H−1π2,+(π2φn)+Kαand Lα∈C00 n−1H−1π2,+(π2φn)+Kα, exchanging C0 n−1and C00 n−1if necessary. Let us study then those other objects, and start with the hypergeometric D-module Hn−1. 20 ALBERTO CASTA˜ NO DOM´ INGUEZ Write Hn−1=Hγ(αi;βj) for certain γ, αi, βjand i, j = 1, . . . , r (we do not need their concrete expression now) and call g:= z(1 −z). Then, we want to know the kernel of ∂z(acting by left multiplication) at DGmz, g−1h∂zi/(P0, P1), where P0=∂z−dnz−wn z(1 −z)Dλand P1=γzwn(1 −z)dn−1 r Y i=1 (Dλ−αi)−λ r Y j=1 (Dλ−βj). Dividing by P0, every element of DGmz, g−1h∂zican be written as a=PN i=0 aiDi λ, with every coefficient ai∈ OGmz, g−1. Moreover, we can take N=r−1. If N≥r, we have that ∂z·a= N X i=0 ∂z(ai)Di λ+ N X i=0 (dnz−wn)g−1aiDi+1 λ, so taking the symbols with respect to Dλ, (dnz−wn)g−1aN=xγzwn(1 −z)dn−1−λ for some x, and so, aN=yγzwn(1 −z)dn−1−λ for some y. Therefore, a=a0+yDN−r λP1, where degDλa0< N and such that ∂zabelongs to the ideal (P0, P1) if and only if ∂za0does. Take then a=Pr−1 i=0 aiDi λ, and let us assume that ∂za∈(P0, P1). As above, there will exist some x and ybelonging to OGmz, g−1such that (dnz−wn)g−1ar−1=xγzwn(1 −z)dn−1−λand ar−1=yγzwn(1 −z)dn−1−λ. Since degDλ∂za=r, we necessarily have that ∂za=xP1. But H−1π2,+(π2φn)+Hn−1is an extension of Kummer D-modules: either it is a subobject of H−1π2,+(π2φn)+Gn−1or it lies in between it and some Kummer D-modules according to the two possibilities in 6, so there must exist some αsuch that (Dλ−α)a=y0P1. Now ∂zaand (Dλ−α)(dnz−wn)g−1ashare their leading term in Dλ, so ∂z−(Dλ−α)(dnz−wn)g−1a will vanish. Therefore, for every i= 0, . . . , r −1 we will have that ∂z(ai)−(dnz−wn)g−1(Dλ−α)(ai)=0. Writing locally each aias Pj∈Zaijλj, with each aij belonging to kz, g−1, we must have that, for any iand j, ∂z(aij) = (dnz−wn)g−1(j−α)aij. This differential equation has as formal solution space the k-span of zwn(1 −z)dn−1α−j, which is algebraic only if wnα, dn−1α∈Z. In conclusion, H−1π2,+(π2φn)+Hn−1is an extension of certain Kα for wnαand dn−1αbeing an integer. Let us turn our attention now to the Kummer side of Gn−1. Thanks to Lemma 4.6, the D-module H−1π2,+(π2φn)+Kαdoes not vanish if and only if wnα, dn−1α∈Z. In conclusion, gathering the information of the last paragraphs, we know that H−1π2,+(π2φn)+Gn−1must be an extension of Kummer D-modules Kαfor some αsuch that wnαand dn−1αare always integer numbers. Point iii of Proposition 3.9 tells us that G−1, being a Kummer D-module Kα, must verify that dnα is an integer. From the second row from the last of the long exact sequence 11 we can affirm that it is also a subobject of either H−1π2,+(π2φn)+Gn−1or Len−1 a=1 K(n−1 n−3) a/en−1. In any case we would have that dn−1α∈Z, so wnαshould be an integer as well. However, G−1is the nonconstant composition factor of H−1Kn(if there is any), and we are imposing a condition on αthat depends heavily on the order we have followed to perform the induction, whereas Kndoes not change when changing the variables and thus the inductive steps. Renaming the variables so that we exchange wnand any other wiand following the same inductive procedure, we obtain that G−1must actually be a nontrivial Kummer D-module Kαsuch that wiα∈Zfor every i= 0, . . . , n. DWORK FAMILIES AND D-MODULES 21 But if such a thing happens, then enαshould also be an integer, and by assumption en= 1, so in the end G−1= 0 and we have finished the proof.  Remark 4.8.By applying formula 9 and looking at the second row from the last of the long exact sequence 11 we can also deduce from the proof of the proposition that H−1π2,+(π2φn)+Gn−1= 0 and H−1π2,+(π2φn)+H0Kn−1∼ =On−1 Gm. Consequently, we have as well an exact sequence which will be very useful in the following result, formed by the last row of the long exact sequence 11: 0−→ H0Kn−→ H0π2,+(π2φn)+Kn−1−→ en−1 M a=1 Kn−2 a/en−1⊕ dn−1 M a=1 Ka/dn−1−→ 0. What we still have to prove is the rest of the properties of Gnthat are of interest to us: its generic rank, its singular points and its exponents at the origin and infinity. The first two can still be obtained within the inductive context. Proposition 4.9. For every n≥1, the generic rank of Gnis dn−en, and it has a unique singularity at Gm, namely at γn. Proof. We already know that Gnis a regular holonomic DGm-module, and its Euler-Poincar´e characteristic is −1, so by Proposition 2.6 it will have a singularity at some point λ0. Its restriction to the rest of Gmwill then be a module with integrable connection of some rank to be determined. Since we know by Lemma 3.10 that the statement of the proposition is true for n= 1, let us prove it for a general nby induction, and so let us assume its veracity for lower values of the index n. Assume as well that en= 1; as before the general case can be easily deduced from this one. As we discussed in Remark 4.8, we had an exact sequence of the form 0−→ H0Kn−→ H0π2,+(π2φn)+Kn−1−→ en−1 M a=1 Kn−2 a/en−1⊕ dn−1 M a=1 Ka/dn−1−→ 0. Consider the long exact sequence of cohomology of the triangle π2,+(π2φn)+Gn−1−→ π2,+(π2φn)+H0Kn−1−→ π2,+(π2φn)+ en−1 M a=1 Kn−1 a/en−1, and focus in degree −1. We know that H−1π2,+(π2φn)+Gn−1vanishes and H−1π2,+(π2φn)+H0Kn−1∼ = On−1 Gmand, by formula 10, that H0π2,+(π2φn)+ en−1 M a=1 Kn−1 a/en−1 ∼ =On−1 Gm[1] ⊕ On−1 Gm⊕ en−1 M a=1 Kn−1 a/en−1. Therefore, we have in fact the exact sequence 0−→ π2,+(π2φn)+Gn−1−→ H0π2,+(π2φn)+H0Kn−1−→ On−1 Gm⊕ en−1 M a=1 Kn−1 a/en−1−→ 0. Summing up, the generic rank of Gnis equal to −dn−1+en−1−1 plus the generic rank of π2,+(π2φn)+Gn−1; let us find that one. Consider (π2φn)+Gn−1. It is a regular holonomic D-module over (Gm−{1})×Gm, having singularities along the curve Cλ:= (π2φn)−1(γn−1) with equation λ=γn−1zwn(1 −z)dn−1. Fixing µin Gm, the intersection of Cλand the line λ=µis a set Cµof dnpoints whenever µ6=γn; it is formed by dn−1 points otherwise. Now, for any value µof λ, consider the cartesian diagram (Gm− {1})× {µ}  π2  id ×iµ //(Gm− {1})×Gm π2  {µ}iµ //Gm . 22 ALBERTO CASTA˜ NO DOM´ INGUEZ By applying base change, i+ µπ2,+(π2φn)+Gn−1∼ =π2,+(id ×iµ)+(π2φn)+Gn−1. Assume that µ6=λ0. Since π2,+(π2φn)+Gn−1has no singularities at µ,i+ µπ2,+(π2φn)+Gn−1is a single k-vector space, and its dimension equals the generic rank of π2,+(π2φn)+Gn−1. On the other hand, the morphism π2φn(id ×iµ) is ´etale, so the complex (id ×iµ)+(π2φn)+Gn−1is actually a single DGm−{1}-module of rank dn−1−en−1. Its Euler-Poincar´e characteristic will be −dim H0π2,+(id ×iµ)+(π2φn)+Gn−1=−dim i+ µπ2,+(π2φn)+Gn−1=−rk π2,+(π2φn)+Gn−1. Recall that Gn−1has no punctual part, so neither (id ×iµ)+(π2φn)+Gn−1has. Then it will be the intermediate extension of its restriction to (Gm− {1})−Cµand applying [Kat90, Thm. 2.9.9], χ(id ×iµ)+(π2φn)+Gn−1=−(dn−1−en−1)− |Cµ|=−(dn+dn−1−en−1), whenever µ6=γn, where the last summand is a consequence of [op. cit., Lem. 2.9.7] applied to Hn−1. In conclusion, the generic rank of Gnmust be dn+dn−1−en−1−dn−1+en−1−1 = dn−1. Now, if the point λ0where Gnhas a singularity were different from γn, then we could do the same process above and see that χi+ µ(π2φn)+Gn−1=−(dn+dn−1−en−1−1), which cannot be possible. Therefore, λ0=γn. 5. Exponents of Gnand proof of the main result In the previous section we proved every part of the main theorem that could be accomplished by the inductive process; we will provide in this one the proof of Theorem 1.5 by using all the information given by the results of section 4 and finding the values for the exponents of Gnat both the origin and infinity. Proof of Theorem 1.5. Assume again that gcd(w0, . . . , wn) = 1. The statement about the constant part of Knand the existence of Gnand the exact sequence 0−→ Gn−→ H0Kn−→ On Gm−→ 0 is Proposition 4.7. Now by Proposition 4.9, Gnis a regular DGm-module of Euler-Poincar´e characteristic −1, of rank dn− 1 and singularities at the origin, γnand infinity, so by Propositions 2.1, 2.5 and 2.6 its semisimplification will consist of kKummer D-modules and an irreducible hypergeometric D-module of type (dn−1− k, dn−1−k), with a singularity at γn, that is to say, Gss n=M α∈A Kα⊕ Hn, where |A|=kand Hnis an irreducible hypergeometric D-module of type (dn−1−k, dn−1−k) of the form Hγn(αi;βj). Since we want to characterize the Kummer D-modules and Hn, we only need, by virtue of Propositions 2.4 and 2.5, to find the exponents of Gnat both zero and infinity. Thus those occurring at both points will determine the Kummer summands, and the rest will be the parameters of Hn(cf. [Kat90, Cor. 3.7.5.2]). By Remark 3.4, ¯ Kncannot have a singularity at the origin. Since j+¯ Kn=ι+ nKnby construction, the dn−1 exponents at the origin of Gncan only be fractions of denominator equal to dn, say with numerators k1, . . . , kdn−1∈ {1, . . . , dn}. In fact, we can deduce from Theorem 1.4 that the exponents both at the origin and infinity should be, up to a parameter bcoming from tensoring with a Kummer D-module Kb/dn, all of the exponents but one of the hypergeometric DGm-module Hγncancel 1 w0 ,...,w0 w0 ,..., 1 wn ,...,wn wn ;1 dn ,...,dn dn. DWORK FAMILIES AND D-MODULES 23 The one missing will be of the form a/dn, because we know that the rank of Gnis dn−1 thanks to Proposition 4.9. Summing up, a list of the exponents (of some representative of them in kin fact) is, respectively at the origin and infinity, Wn:= j wi +b dn :i= 0, . . . , n, j = 1, . . . , wi−a+b dn, k dn :k= 1, . . . , dn−a+b dn. Now we ought to see which possible values of aand bcan really occur. We will see that we can take a and bto be both of them dnby using the work done at [CD17, §4]. From [op. cit., Thm. 1.2] we know that the exponents at the origin of H0λn,+OZncan only be those of the form j/wimod Z, and each of them which is not integer occurs with the same multiplicity (without counting coincidences among some j/wifor different values of jand i). Let us show that the multiplicity at the origin of one is nat least. Indeed, assume that some wiis strictly bigger than 1 and that the fixed multiplicity for each noninteger expression j/wiis some m > 1. Then there will exist a η0=j0/wi0for some admissible choice of i0∈ {0, . . . , n}and j0∈ {1, . . . , wi0} such that there will be mdifferent admissible pairs (i1, j1),...,(im, jm) with η0=jk/wik+b/dnfor each k= 1, . . . , m. But then this implies that all those jk/wikare the same number, so we must be able to find an exponent η1=j1/wi1with multiplicity m2. Therefore, we can find m2more admissible pairs of the form (i0 1, j0 1),...,(i0 m2, j0 m2) such that η1=j0 k/wi0 k+b/dnfor each k= 1, . . . , m2. As a consequence, all of the j0 k/wi0 kcoincide and so there must be an exponent η2=j0 1/wi0 1of multiplicity m3. If we continue this argument, we must reach a point in which there are no more new admissible pairs of the form (i, j), so since the exponents are not all of them equal (because wi>1 for certain i= 0, . . . , n), necessarily η0=ηlfor some l > 0, but this is a contradiction since they had different multiplicities. In conclusion, either the multiplicity of each noninteger exponent is one, so that the multiplicity of one is n, or the only exponent is one and so its multiplicity is dn−1. That implies that there is no possible choice of aand bother than a=b=dn. Indeed, if a6=dn, then the value b/dnappears at Wnwith multiplicity n+ 1, and if it is not 1, then there must exist n+ 1 different jk< wksuch that jk/wk=jl/wl=b/dn, but this is impossible because gcd(wi) = 1 (cf. [CD17, Lem. 4.2]). Therefore, b=dn, but in that case, since a6=dn, there would exist an exponent at both the origin and infinity equal to 1, and then Gnwould have a composition factor equal to OGm, which cannot happen by Proposition 4.7. Thus a=dn. Then if b6=dnthere must exist again an exponent at both the origin and infinity equal to 1, because its multiplicity at both points is positive, but as before this is a contradiction, so in conclusion a=b=dn. Once we have proved everything above for en= 1, the statement for a bigger enfollows easily taking into account that the direct image [en]+of a irreducible hypergeometric DGm-module remains both irreducible and hypergeometric and the exponents of the direct image are all the classes of k/Zsuch that multiplying by enwe recover the original ones. With this last step we can finish the inductive step and use it in the results of section 4. Last, the exact sequence of Gnis corollary 4.4, once we have proved everything else in Theorem 1.5 about Krfor r=n, n + 1.  Remark 5.1.One could wonder when Knor ¯ Knhave integer exponents (or, equivalently in the complex analytical setting, unipotent formal local monodromy) at the origin and infinity, respectively. This is a remarkable property (and even rare, in the case of Kn), and can be characterized in terms of the wi. The proofs can be found, respectively, in [Kat09, Lem. 8.8] and [DS04, Rem. 2.2]. 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E-mail address:[email protected] Fak¨ ultat f¨ ur Mathematik, Technische Universit¨ at Chemnitz. 09017 Chemnitz (Germany)