Irregular Hodge filtration of some confluent hypergeometric systems

IRREGULAR HODGE FILTRATION OF SOME CONFLUENT HYPERGEOMETRIC SYSTEMS ALBERTO CASTA˜ NO DOM´ INGUEZ AND CHRISTIAN SEVENHECK Abstract. We determine the irregular Hodge filtration, as introduced by Sabbah, for the purely irregular hypergeometric D-modules. We obtain in particular a formula for the irregular Hodge numbers of these systems. We use the reduction of hypergeometric systems from GKZ-systems as well as comparison results to Gauß-Manin systems of Laurent polynomials via Fourier-Laplace and Radon transformations. 1. Introduction The aim of this paper is to compute some Hodge theoretic invariants of certain classical differential systems in one variable. These are the so-called irregular Hodge numbers, which have been introduced recently by Sabbah [Sab18a]. They are called irregular because they are attached to differential systems which may have irregular singular points, a feature that is excluded for classical variations of Hodge structures as well as for the more general Hodge modules. The very definition of these numbers rely on the theory of mixed twistor D-modules of T. Mochizuki ([Moc15a]). Twistor D-modules generalise Hodge modules, in the sense that the underlying D-module of a twistor D-module can have irregular singularities. In particular, one can define a version of the Fourier-Laplace transformation functor within the category of mixed twistor D-modules, which is impossible for mixed Hodge modules. The drawback of this generalization is that one cannot directly assign a filtration to a twistor D-module, and hence it is not easy to attach numerical invariants like Hodge numbers to it. In the above mentioned paper [Sab18a], Sabbah constructs an intermediate category between mixed Hodge modules and mixed twistor D-modules (called irregular mixed Hodge modules) which is on the one hand sufficiently large to be stable under all relevant operations that are defined for twistor D-modules (in particular, the Fourier-Laplace transformation), but which allows one to define a filtration, called irregular Hodge filtration, for each of its objects. The construction is related to the earlier papers [Yu14,ESY17,SY15], where for certain projective morphisms f:X→P1, a (rationally indexed) filtration was introduced on the twisted de Rham complex (Ω•(∗D), d +df∧), where D⊂Xis a certain normal crossing boundary divisor such that f|X\Dyields a regular function. Instead of considering meromorphic differential forms, one can also use the subcomplex of so-called f-adapted logarithmic forms (also called Kontsevich complex, see [KKP17]) Ω• f, which consists of forms such that the exterior product with df is still logarithmic along D. From an E1-degeneration property of the corresponding spectral sequence proved in [ESY17], one obtains a filtration on the twisted de Rham cohomology, called irregular Hodge filtration of the regular function f:X\D→A1. We refer to [ESY17] for more details. Notice also that the paper [KKP17] gives three definitions of so-called Landau-Ginzburg Hodge numbers associated to a family f:X→P1, one of them being dim Hp(Ωq f). Conjecturally these three definitions coincide, but this seems to require some more assumptions (see [Sha18] and [Sab18b] for some partial results). Although these Hodge numbers have only integer indices, they are closely related to the dimensions of the graded parts of the filtration from [ESY17]. Ultimately, according to [KKP17] and following predictions from homological mirror symmetry, one hopes for a correspondence between the Hodge numbers of some, say, projective varieties entering in the A-model and the irregular Hodge numbers of its (Landau-Ginzburg) B-model. 2010 Mathematics Subject Classification. Primary 14F10, 32C38. Key words and phrases. D-modules, irregular Hodge filtration, hypergeometric systems, twistor D-modules. The authors are partially supported by the project SISYPH: ANR-13-IS01-0001-01/02 and DFG grant SE 1114/5-1. 1 2 ALBERTO CASTA˜ NO DOM´ INGUEZ AND CHRISTIAN SEVENHECK For the moment, there are quite a few examples where the irregular Hodge filtration can actually be computed. A central result of [Sab18a] states that rigid irreducible D-modules on the projective line underlie objects of the category of irregular mixed Hodge modules, and consequently admit a unique irregular Hodge filtration, provided that their formal local monodromies are unitary. Rigid D- modules are particularly interesting since they can be algorithmically constructed from simple objects by an algorithm due to Arinkin and Katz (cf. [Ari10]). Among the most studied and best understood examples of such rigid D-modules are the classical hypergeometric D-modules. In the regular case (which corresponds to classical variations of complex Hodge structures), Fedorov has given in [Fed18] a closed formula for the Hodge numbers (without computing the Hodge filtration itself however) conjectured by Corti and Golyshev in [CG11] using the work [DS13] of Dettweiler and Sabbah. In the present paper, we consider the case of purely irregular hypergeometric modules. Our principal result, Theorem 4.7, completely determines the irregular Hodge filtration and gives a very simple formula for the irregular Hodge numbers, which is in some sense similar to the shape of Fedorov’s formula. The main ingredients are a reduction process (as explained in [BMW18]) to obtain classical hypergeometric D-modules from some higher dimensional ones, the so-called GKZ-systems, techniques from [Rei14] and [RS15b, RS17] (going back to [Sab08]) to understand Hodge module structures on these GKZ-systems as well as a quite explicit solution to the so-called Birkhoff problem that is inspired by calculations in toric mirror symmetry (see again [RS15b] as well as [DS04] and also [GMS09]). Since the first version of this paper was written, Hodge invariants for hypergeometric systems have been considered in some other articles. First, the formula for the Hodge numbers of purely irregular systems has also been obtained by Sabbah and Yu in the final version of [Sab18a] by different means. Another approach to Fedorov’s formula for the case of regular systems (giving more precise information on the various Hodge invariants) has been given by N. Martin (see [Mar18]). Moreover, we have considered the case of hypergeometric operators of type (n, 1) (see Definition 2.1 below) in a common paper with Th. Reichelt ([CDRS18]) using the computation of Hodge filtrations on GKZ-systems from [RS15a]. Finally, Sabbah and Yu have recently given a complete formula for the irregular Hodge numbers for all confluent hypergeometric systems in [SY18]. However, for the moment the irregular Hodge filtration itself is not determined in the general case. Let us recall some notation that will be adopted throughout the paper. For a smooth complex algebraic variety X, we write DXfor the sheaf of algebraic differential operators on X. If Xis affine, we sometimes switch freely between sheaves of DX-modules and modules of global sections. We will denote the abelian categories of holonomic resp. regular holonomic DX-modules by Modh(DX) resp. Modrh(DX), and analogously with the respective bounded derived categories. For a morphism f:X→Y, we denote the direct resp. inverse image functors for D-modules as usual by f+resp. f+ (see [HTT08] for a thorough discussion of these and related functors). We put Gm:= Spec C[t, t−1]; if we want to fix a coordinate on this one-dimensional torus, we also write Gm,t. We denote by Rint A1 z×Xthe sheaf of Rees rings on X(with integrable structure), that is, the subsheaf of non-commutative algebras of DA1 z×Xgenerated by zp∗ΘXand z2∂z, where p:A1 z×X→ Xis the projection. If (x1, . . . , xn) are local coordinates on X, then Rint A1 z×Xis locally given by OA1 z×Xhz2∂z, z∂x1, . . . , z∂xni. Occasionally, we will also need the sheaf RA1 z×X, which is generated by zp∗ΘXonly, i.e., locally given by OA1 z×Xhz∂x1, . . . , z∂xni. We let MHM(X) be the abelian category of algebraic mixed Hodge modules (see [Sai90]) on X. We consider moreover the category MHM(X, C) of complex mixed Hodge modules (see, e.g., [DS13, Definition 3.2.1]). As an example, if Xis the algebraic torus Gd mwith coordinates y1, . . . , yd, then for any α= (α1, . . . , αd)∈Rd, the free OGd m-module of rank 1 Oα Gd m:= DGd m/(yk∂yk+αk+ 1)k=1,...,d (see also Definition 3.7 below) underlies an object in MHM(Gd m,C). On the other hand, MTM(X) denotes the abelian category of algebraic mixed twistor D-modules on X(see [Moc15a]). The category MTMint(X) consists of those mixed twistor D-modules where the underlying RA1 z×X-modules have an integrable structure, i.e., where they are modules over Rint A1 z×X. The category IrrMHM(X), as defined in [Sab18a, Def. 2.52], is a certain subcategory of MTMint(X) (actually of a variant, called ιMTMint(X), which is shown to be equivalent to MTMint(X) in [ibid., IRREGULAR HODGE FILTRATION OF SOME CONFLUENT HYPERGEOMETRIC SYSTEMS 3 §1]) consisting of objects c Mthat satisfy certain properties. The first of them is that the object θc M obtained from c Mby substituting zθ for zis still an object of ιMTMint(θX), where θX=Gm,θ ×X. That is a remarkable assumption, having its origin in [HS07], but we have to impose a further property, namely that such rescaled objects must have a certain tame behaviour when θgoes to infinity (or at the origin of τ= 1/θ, in other words), related to regularity along {τ= 0} × A1 z×X. This two conditions are, essentially, what we ask for a mixed twistor D-module on Xto be a mixed twistorrescaled D-module on X([Sab18a, Def. 2.50]). The last condition is a z-grading property appearing after identifying τwith z(cf. [ibid., Def. 2.26]). For more details, see [ibid., §2]. Although the construction of this category may seem rather involved, its main feature is that the DX-module Massociated to an object c Min IrrMHM(X) carries a good filtration Firr •M, indexed by R, called the irregular Hodge filtration, which in turn behaves well with respect to several functorial operations. Notice however that this filtration, contrarily to the case of mixed Hodge modules, is not part of the definition of an object of IrrMHM(X). Very roughly, it can be thought of as defined by the intersection of the canonical V-filtration along τ= 0 (or rather the filtration induced on the restricted object when τ=z) with the z-adic filtration on c M. In particular, the jumping indices of the irregular Hodge filtration are of the form {α+k|k∈Z}for a certain finite set of real numbers α. Acknowledgements. We would like to thank Takuro Mochizuki, Thomas Reichelt and Claude Sabbah for their interest in our work and for many stimulating discussions. We are grateful to the anonymous referee for the many valuable comments and remarks. We thank the Max Planck Institute for Mathematics in the Sciences, where a significant part of the work presented here has been carried out. 2. Hypergeometric modules and dimensional reductions In this section we will introduce two different kinds of hypergeometric D-modules: classical and GKZ. We will dedicate more time to classical ones, since they form one of the main objects of study of the paper, and will end by showing the relation between both types, which will be useful for us in the future. We state one of our main results (Theorem 2.13), which is proven in the next section. Definition 2.1. Let (n, m)6= (0,0) be a pair of nonnegative integers, and let α1, . . . , αnand β1, . . . , βm be elements of C. The (classical) hypergeometric D-module of type (n, m) associated with the αiand the βjis defined as the quotient of DGmby the left ideal generated by the so-called hypergeometric operator n Y i=1 (t∂t−αi)−t m Y j=1 (t∂t−βj). We will denote it by H(α1, . . . , αn;β1, . . . , βm), or in an abridged way, H(αi;βj). In this paper we will be mostly concerned with hypergeometric D-modules of type (n, 0). Remark 2.2.The excluded type (n, m) = (0,0) corresponds to the punctual delta DGm-module on Gm H(∅;∅) = DGm/(1 −t). On the other hand, if we denote the Kummer D-module DGm/(t∂t−η) by Kη, for any fixed complex number η, then H(αi;βj)⊗OGmKη∼ =H(αi+η;βj+η). In particular, an overall integer shift of the parameters gives us an isomorphic D-module. Every hypergeometric D-module has Euler characteristic -1 (cf. [Kat90, Lem. 2.9.13]). For n6=m, no hypergeometric D-module of type (n, m) has singularities on Gm. If n>m(resp. m>n), they have a regular singularity at the origin (resp. infinity) and an irregular singularity at infinity (resp. the origin) of irregularity one and slope 1/|n−m|of multiplicity |n−m|(cf. [Kat90, Prop. 2.11.9]). For n=m, the hypergeometric D-modules of type (n, n) are regular, with singularities only at the origin, infinity and 1. Proposition 2.3 (Irreducibility).(cf. [Kat90, Prop. 2.11.9, 3.2]) Let H:= H(αi;βj)be a hypergeometric D-module. It is irreducible if and only if for any pair (i, j)of indices, αi−βjis not an integer. 4 ALBERTO CASTA˜ NO DOM´ INGUEZ AND CHRISTIAN SEVENHECK Remark 2.4.Assuming the irreducibility, we have a result which is stronger than the second paragraph of Remark 2.2. Namely, the isomorphism class of H(αi;βj) depends only on the classes modulo Zof the αiand the βj(point 1 of [Kat90, Prop. 3.2]), so we can choose such parameters on a fundamental domain of C/Z. Proposition 2.5 (Rigidity).Let H:= H(αi;βj)be an irreducible hypergeometric D-module of type (n, m), where n≥m, and let Mbe another irreducible DGm-module of Euler-Poincar´e characteristic -1 which has no singularities outside {0,1,∞}. Assume that • H ⊗ C((t)) ∼ =M ⊗ C((t)). • H ⊗ C((1/t)) ∼ =M ⊗ C((1/t)). •If n=m, assume further that Mhas a regular singularity at 1. In that case, Hand Mare isomorphic. Proof. Hand Mbeing irreducible, both of them coincide with the middle extension of their restriction to Gm\{1}. Assume first that n=m, that is, His regular. Since Mhas characteristic -1 and regular singularities at the origin, one and infinity, by the formula for the Euler characteristic [Kat90, Thm. 2.9.9], its formal local monodromy at 1 must be a pseudoreflection. Then we can apply [ibid., Rigidity Thm. 3.5.4] and we are done. If n>m,M, like H, has a regular singularity at the origin and an irregular singularity at infinity. By the same formula for the Euler characteristic of j†+j+M(denoting by jthe inclusion Gm\{1},→Gm), Mcannot have more singularities in P1apart from zero and infinity. In that case we apply [ibid., Rigidity Thm. bis 3.7.3].  Consider the case in which H=H(αi;βj) is of type (n, n), i.e., such that it is regular. From the rigidity property of the last Proposition, one concludes by [Sim90, Cor. 8.1] that the restriction of Hto Gm\{1}underlies a complex polarizable variation of Hodge structures. The next statement, conjectured by Corti and Golyshev (cf. [CG11, Conj. 1.4]) and proved by Fedorov (see [Fed18, Thm. 1]), gives important information on its Hodge invariants. Proposition 2.6 (Hodge numbers (regular case)).Let H=H(αi;βj)be an irreducible hypergeometric D-module of type (n, n). Assume that the αiand the βjare increasingly ordered real numbers, lying in the interval [0,1). Set ρ(k) = |{i= 1, . . . , n :βi< αk}| − k, for k= 1, . . . , n. Then the Hodge numbers of Hare, up to an overall shift, hp=ρ−1(p)=|{k= 1, . . . , h :ρ(k) = p}| . This last result is the analogous one, in the regular case, to Theorem 4.7, and served as the main motivation to start this project. Notice that this formula slightly differs from the one given in [Fed18]. This is because Fedorov considers the dual connection to ours by working with the space of solutions instead of with that of horizontal sections (cf. [Mar18, p. 10, proof of Lem. 3.6]). Now that we have seen part of the behaviour of classical hypergeometric D-modules, let us continue with the other family mentioned above. Definition 2.7. Let n≥mtwo positive integers, and let d=n−m. Let β∈Cdbe a vector and let A= (aij)∈M(d×n, Z) be an integer matrix. Consider the n-dimensional torus Gn mwith coordinates λ1, . . . , λn. We define the Euler operators Ei=Pjaijλj∂λj, for i= 1, . . . , d, and the toric ideal IA:= (∂u λ−∂v λ:Au =Av)⊂C[∂λ1, . . . , ∂λn]⊂DAn. Then, the GKZ-hypergeometric D-module (or system) is Mβ A:= DGn m/(IA+ (Ei−βi:i= 1, . . . , d)i=1,...,d), where IAis the sheafified version of the toric ideal IA. Usually the module Mβ Ais defined to be an element in Mod(DAn). However, we will later work only with the restriction of such an object to Gn m. In order to avoid the usage of the functor j+(where j:Gn m,→An) each time that we need to refer to Mβ A, we use this slightly non-standard definition. IRREGULAR HODGE FILTRATION OF SOME CONFLUENT HYPERGEOMETRIC SYSTEMS 5 A classical hypergeometric D-module can be considered as a dimensional reduction of a certain GKZ-system. We will describe this procedure in some more detail now, because it allows us to apply some of the many known results on GKZ-systems to classical hypergeometric D-modules. Proposition 2.8. Let mbe a positive integer. Let A∈M((m−1) ×m, Z)be an integer matrix of rank m−1and take κ∈Cmsuch that κ1= 0. Consider the inclusion ι:Gm,→Gm mgiven by t7→ (t, 1,...,1), and let B= (b1, . . . , bm)t∈Zmbe a Gale dual of A, that is, an integer column matrix which generates kerQA. Assume moreover that b1= 1. Put η:= m Y i=1 bbi i. Then we have h+ ηH(αi;βj)∼ =ι+MAκ A, where hηis the automorphism of Gmgiven by t7→ ηt, and the unordered sets of parameters αiand βj, counted with multiplicities, are (αi) = k−κj bj :bj>0, k = 0, . . . , bj−1, (βi) = k−κj bj :bj<0, k = 0,...,−bj−1. Proof. On one hand, MAκ Ais not only a GKZ-hypergeometric D-module, but also the restriction to Gn mof a lattice basis binomial DAm-module (cf. [BMW18, Def. 1.2], noting that the assumption on the columns of Bis not needed for the definition). This is because Abeing of rank m−1 implies that the toric ideal IAcoincides with the lattice basis ideal I(B) := ∂w+ λ−∂w− λ:w=w+−w−is a column of B. (In fact this holds for any complete intersection ideal, but here the argument is simpler.) On the other hand, the expression we have given for the parameters of H(αi;βj) follows from applying the definition of Horn hypergeometric D-modules given in [ibid., Def. 1.1] for a column matrix, up to the same caveat above about the columns on B(in fact normalized ones, but all kinds of Horn D-modules defined in loc. cit. are equal once restricted to the torus Gm,t), and comparing it with Definition 2.1. Now let j:Gm,→A1be the canonical inclusion. The isomorphism given in [ibid., Thm. 1.4] relates lattice basis binomial D-modules to Horn hypergeometric ones. However, due to the previous discussions, we obtain the isomorphism in the statement just by applying j+to both sides of it.  Note that the choice of κ1and b1in the statement of the Proposition force α1to vanish. However, by Remark 2.2, any other hypergeometric D-module can be built from one of this form just by tensoring with a suitable Kummer D-module. In our study of Hodge theoretic properties of hypergeometric DGm-modules below, we need to go in some sense in the opposite direction: Given sets {αi},{βj}, we would like to determine a matrix A and a parameter vector βsuch that the module H(αi;βj) can be obtained as an inverse image of the GKZ-system Mβ A. Although such a pair (A, β) is not unique, a systematic way of constructing it can be formulated as follows. Corollary 2.9. Let H(αi;βj)be a hypergeometric DGm-module of type (n, m)with n > 0and α1= 0. Let A∈M((m+n−1) ×(n+m),Z)given by A= 1m0m×(n−1) Idm 1n−1−Idn−10(n−1)×m!, and let β= (β1, . . . , βm, α2, . . . , αn)t. (Here and later in this paper, we write xkfor a column vector with krows, where all entries contain the value x.) Let ι:Gm→Gn+m m, given by t7→ (t, 1. . . , 1).Then H(αi;βj)∼ =ι+Mβ A. Proof. The statement is an easy consequence of the Proposition, taking B= (1,(n) . . ., 1,−1,(m) . . ., −1)t and κ= (0,−α2,...,−αn, β1, . . . , βm)t. 6 ALBERTO CASTA˜ NO DOM´ INGUEZ AND CHRISTIAN SEVENHECK As indicated before, we will see later that the restriction α1= 0 is not as strong as it may appear: By tensoring a given hypergeometric D-module with an appropriate Kummer module, we can always reach this assumption. We will end this section by explaining how the above construction of GKZ-systems and the dimensional reduction to hypergeometric DGm,t -modules can be understood at the level of R-modules. Recall (see the introduction) that for a smooth algebraic variety Xwith local coordinates (x1, . . . , xn) the sheaf Rint A1 z×Xis the subsheaf of DA1 z×Xlocally generated by z2∂zand (z∂xi)i=1,...,n. Definition 2.10. Let n≥mtwo positive integers, and let d=n−m. Let β∈Cdbe a vector and let A= (aij)∈M(d×n, Z) be an integer matrix. Consider the affine space Gn mwith coordinates λ1, . . . , λn, and let L⊂Znbe the kernel of the linear map Zn→Zdgiven by left multiplication by the matrix A, whose elements will be denoted by l= (l1, . . . , ln). Then, the GKZ-hypergeometric R-module is c M(β0,β) A:= Rint A1 z×Gn m/I, where β0∈Cand Iis generated by Y j:lj>0 (z∂λj)lj−Y j:lj<0 (z∂λj)−lj, l ∈L, z2∂z+λ1z∂λ1+. . . +λnz∂λn−zβ0, n X j=1 akjλjz∂λj−zβk, k = 1, . . . , d. Note that we can recover the GKZ-hypergeometric D-module Mβ Afrom Definition 2.7 by restricting c M(β0,β) Ato z= 1. In the special case of our original matrix from Corollary 2.9 the generators of Iare (z∂λ1)·. . . ·(z∂λn)−(z∂λn+1 )·. . . ·(z∂λn+m), z2∂z+λ1z∂λ1+. . . +λn+mz∂λn+m−zβ0, λ1z∂λ1+λn+iz∂λn+i−zβi, i = 1, . . . , m, λ1z∂λ1−λiz∂λi+zαi, i = 2, . . . , n. Moreover, we must also consider the corresponding R-module for hypergeometric D-modules. Both kinds of R-modules will play a significant role in the proof of Theorem 2.13. Definition 2.11. Let (n, m)6= (0,0) be a pair of natural numbers, and let α1, . . . , αnand β1, . . . , βm be elements of C. The (classical) hypergeometric R-module (of type (n, m)) associated with the αi and the βj, denoted by b H(αi;βj), is defined as the quotient of Rint A1 z×Gm,t by the left ideal generated by P=z2∂z+ (n−m)tz∂t+γz and H= n Y i=1 z(t∂t−αi)−t m Y j=1 z(t∂t−βj), where γ=−Pn i=1 αi+Pm j=1 βj. The choice of the operator Pmay seem odd, but as we will see, it is indeed very natural. In fact, we have the following extension of Corollary 2.9 to the realm of R-modules. Lemma 2.12. Let b H(αi;βj)be a classical hypergeometric RA1 z×Gm,t -module of type (n, m)with n > 0 and α1= 0. Let A∈M((m+n−1) ×(n+m),Z),β∈Cn+m−1and ι:Gm,t ,→Gn+m mbe as in the statement of Corollary 2.9. Then b H(αi;βj)∼ =ι+c M(0,β) A. Proof. The inverse image functor in the category of R-modules is induced by the usual inverse image functor of O-modules, (idA1 z×ι)∗in this case (cf. [Moc15a, §2.1.6.2]). Then it is easy to see that b H(αi;βj)∼ =ι+c M(0,β) A. IRREGULAR HODGE FILTRATION OF SOME CONFLUENT HYPERGEOMETRIC SYSTEMS 7 Namely, we replace zλi∂λiby zλ1∂λ1−zαiif i= 2, . . . , n or by −zλ1∂λ1+zβi−n, if i=n+1, . . . , n+m. Since we can invert λiin Rint A1 z×Gn m, we present c M(0,β) Aas the OGn+m m-module OGn+m mhz2∂z, zλ1∂λ1i/J, where Jis generated by λn+1 ·. . . ·λn+m n Y i=1 z(λ1∂λ1−αi)−(−1)mλ1·. . . ·λn m Y i=1 z(λ1∂λ1−βi) and z2∂z+ (n−m)zλ1∂λ1+γz. Now the inverse image by ιamounts simply to set λ1=tand λi= 1 for i= 2, . . . , n +min the generators of the ideal, from which the desired isomorphism follows, up to multiplying tby −1.  We can formulate at this point one of the main results of this paper. Its full proof will occupy the entire next section. Theorem 2.13. Let α1, . . . , αnbe real numbers, belonging to the interval [0,1) and increasingly ordered, and let γ=−Pn i=1 αi. Then the RA1 z×Gm,t -module b H:= b H(αi,∅) = Rint A1 z×Gm,t /(P, H), where P=z2∂z+ntz∂t+γz and H= n Y i=1 z(t∂t−αi)−t, underlies an irregular Hodge module, i.e., an object of IrrMHM(Gm,t). It is the unique irregular Hodge module whose associated DGm,t -module is H(αi;∅). Moreover, b Hcan be extended in a unique way to an Rint A1 z×P1-module, b Hpr, such that it underlies an object of IrrMHM P1. Proof of unicity. We will give here a proof of the two unicity statements in the above theorem, postponing the proof of the main statement to page 25 below. Consider any twistor D-module b H0on Gm,t whose underlying DGm,t -module is H. Since the functor ΞDR is faithful by [Moc15a, Rem. 7.2.9], we have an injection of Hom groups HomMTM(Gm,t)(b H,b H0),→HomDGm,t (H,H). But His irreducible, so its only endomorphism is the identity and then, a twistor D-module underlying His unique, if it exists. On the other hand, let j:Gm,t ,→P1be the canonical inclusion and consider the DP1-module Hpr := j†+H. It is an irreducible holonomic DP1-module, because so is Hby Proposition 2.3. Then it gives rise to a unique pure integrable twistor D-module b Hpr on P1by [Moc11, Thm. 1.4.4] and [Sab18a, Rem. 1.40]. In addition, its underlying DP1-module Hpr is rigid by virtue of Proposition 2.5. As a consequence, we can invoke [Sab18a, Thm. 0.7] and claim that such twistor D-module on P1is in fact an object of IrrMHM(P1). Take now b H0:= j+b Hpr, which is an irregular mixed Hodge module whose underlying DGm,t -module is H, by [Moc15a, Prop. 14.1.24]. The we must have, as was just shown, b H0∼ =b H, so that the extension b Hpr of b His unique, as claimed.  The main point in the above theorem is that b Hunderlies an irregular mixed Hodge module. Since the proof of this fact is rather long, and will be carried out in the next section through various intermediate results, we would like to orient the reader by giving here an overview of these steps. We will restrict the sketch to the case where α1= 0, this is also the first (and main) step in the actual proof below on page 25. The general case can be rather easily deduced from this special one by considering Kummer D- resp. R-modules. The first point is to realise b Hin a geometric way. For this purpose, consider the following two families of Laurent polynomials f(y1, . . . , yn−1, λ1, . . . , λn) := −λ1·y1·. . . ·yn−1−λ2 y1 −. . . −λn yn−1 and 0 f(y1, . . . , yn−1, t) := −t·y1·. . . ·yn−1−1 y1 −. . . −1 yn−1 . 8 ALBERTO CASTA˜ NO DOM´ INGUEZ AND CHRISTIAN SEVENHECK where yk, λi, t ∈Gm. Write ι:Gm,t ,→Gn m,t7→ (t, 1,...,1), so that we have the cartesian diagram Gn−1 m×Gm,t Gn−1 m×Gn m A1 λ0×Gm,t A1 λ0×Gn m 0 f×π2f×π2 idA1 λ0 ×ι where we denote the coordinate on the affine line corresponding to the value of fresp. of 0 fby λ0, for reasons that will become clear later. We consider the so-called twisted cohomology groups associated to the morphisms fand 0 f. It can be shown (see Proposition 3.8 below) that c M(0,α) A∼ =Hn−1π2,∗Ω•+d Gn−1 m×Gn m/Gn m [z], z (d−κ(α)∧)−df∧, where (0, α) = (0, α1, α2, . . . , αn) = (0,0, α2, . . . , αn), where κ(α) = Pn−1 j=1 αj+1dyj/yjand Ais the matrix from Corollary 2.9 for the case m= 0. Moreover, since the twisted cohomology groups involve complexes of relative differential forms, we have ι+Hn−1π2,∗Ω•+d Gn−1 m×Gn m/Gn m [z], z (d−κ(α)∧)−df∧ ∼ =Hn−1π2,∗Ω• Gn−1 m×Gm,t/Gm,t [z], z (d−κ(α)∧)−d0 f∧, so that by using Lemma 2.12 we obtain an isomorphism of Rint A1 z×Gm,t -modules (1) b H∼ =Hn−1π2,∗Ω• Gn−1 m×Gm,t/Gm,t [z], z (d−κ(α)∧)−d0 f∧. As a second step, we will realize the right hand side of the above isomorphism in a different way. Namely, write 0 ϕ:= ( 0 f, π2) : Gn−1 m×Gm,t →A1 λ0×Gm,t and consider the direct image complex 0 ϕ+Oα Gn−1 m∈Db rh(DA1 λ0×Gm,t ), where Oα Gn−1 m=DGn−1 m/(yj∂yj+αj+1 + 1)j=1,...,n−1. We are interested in the top cohomology of this complex. Standard techniques for the calculation of direct images of D-modules show that it is given by M:= π2,∗Ωn−1 Gn−1 m×Gm,t/Gm,t [∂λ0] (d−∂λ0·d0 f∧)π2,∗Ωn−2 Gn−1 m×Gm,t/Gm,t [∂λ0] We will use a variant of the Fourier-Laplace transformation (called localized partial Fourier-Laplace transformation, and denoted by FLloc Gm,t , see Definition 3.10 and Definition 3.11 below) exchanging the operator ∂λ0·into z−1·, the operator λ0·into z2∂z·, and localizing along z=∞. Then we have an isomorphism of DA1 z×Gm,t -modules FLloc Gm,t (M) = π2,∗Ωn−1 Gn−1 m×Gm,t/Gm,t [z±] (z·d−d0 f∧)π2,∗Ωn−2 Gn−1 m×Gm,t/Gm,t [z±] ∼ =b H[z±]⊃b H. One of the main points of the proof in the next section is to give a good description of the image of b Hinside FLloc Gm,t (M) under this isomorphism. One such description is given by the isomorphism of Rint A1 z×Gm,t -modules in the displayed formula (1). However, we cannot, a priori, obtain any Hodge theoretic information on b Hfrom (1). On the other hand, we know that Munderlies an algebraic complex mixed Hodge module on A1 λ0×Gm,t (since it is the direct image of such an object on Gn−1 m), and hence it comes equipped with a certain good filtration FH •M(the Hodge filtration). There is a general procedure, explained below in Definition 3.20 and Lemma 3.21, which constructs, given a IRREGULAR HODGE FILTRATION OF SOME CONFLUENT HYPERGEOMETRIC SYSTEMS 9 filtered DAλ0×Gm,t -module (N, F•), a Rint A1 z×Gm,t -module called GF 0FLloc Gm,t Nsuch that its localisation GF 0FLloc Gm,t N⊗OA1 z×Gm,t OA1 z×Gm,t [z−1] equals FLloc Gm,t N. Then we show in Theorem 3.24 that b H∼ =GFH 0FLloc Gm,t M up to a shift of the Hodge filtration. Actually, the proof is not that direct, since we have to identify FLloc Gm,t (M) with the localized partial Fourier-Laplace transformation of some other DA1 λ0×Gm,t -module (called M†+), which underlies a pure polarizable Hodge module. It is constructed by taking a compactification of 0 f, i.e., a projective morphism defined on a quasi-projective (usually singular) variety constructed from the toric compactification of Gn−1 minside Pn. Then M†+is obtained as the direct image under this projective morphism of a certain intersection cohomology module. Now it is known (see [Sab18a, Cor. 0.5]) that if M†+underlies a pure polarizable Hodge module, the Rint A1 z×Gm,t -module GFH 0FLloc Gm,t M†+underlies an irregular Hodge module, which finishes the proof. Notice that the proof of the identification FLloc Gm,t (M)∼ =FLloc Gm,t (M†+) is derived from a similar isomorphism for the direct images of the morphisms ϕ= (f, idGn m) resp. its compactification, rather than for 0 ϕ, and is done via the formalism of Radon transformations for regular holonomic D-modules (in the same way as in [Rei14] and [RS17,RS15a]). 3. Hodge modules and Fourier-Laplace transformation Let α1, . . . , αnbe real numbers, and consider the hypergeometric DGm-module H=H(αi;∅). As we have mentioned before, the goal of this section is to prove Theorem 2.13 above, showing that the R-module b H:= b H(αi;∅) from Definition 2.11 underlies an object of IrrMHM(Gm) (the abelian category of exponential Hodge modules as defined in [Sab18a], see the introduction). We have already indicated several times that we will use the GKZ-hypergeometric R-module c M(0,α) Afor the matrix A from corollary 2.9, but where m= 0. However, we will start with a more general situation, and specify the assumptions we need when going on with the proof. Let d < n be two positive integers, and take a parameter vector β∈Cd. For the Hodge theoretic questions we are interested in, only real parameter vectors are relevant, but we will work with this more general setting until Remark 3.17 below. Let A= (a1, . . . , an)∈M(d×n, Z) be an integer matrix satisfying the following: Assumptions 3.1. (i) ZA=Zd, here ZA:= Pn i=1 Zai, (ii) Let ∆ := Conv(a1, . . . , an) be the convex hull in Rdof the vectors given by the columns of the matrix A. Then for any proper face Γ ⊂∆ we require that the set {ai|ai∈Γ}is part of a Q-basis of Qd. (iii) The origin lies in the interior of ∆. Remark 3.2.(i) These assumptions are in particular satisfied if a1, . . . , anare the primitive integral generators of the rays of the fan Σ defining a toric Fano orbifold XΣ, since in this case it is known (see [CK99, Lem. 3.2.1 and §3.5]) that Σ is the union of the cones over the proper faces of ∆, so that assumption (ii) is satisfied by the fact that the cones of Σ are simplicial. Moreover, for Fano toric varieties the origin is the only integer point in the interior of ∆, so (iii) obviously holds. In any case, we see that these conditions are satisfied for the example (2) A=     1−100. . . 0 1 0 −1 0 . . . 0 . . .. . . 1 0 0 0 . . . −1     ∈M ((n−1) ×n, Z), since these are the generators of the rays of the fan of Pn−1. As we have seen in Corollary 2.9, this is the matrix we have to look at when we want to express the classical hypergeometric D-module of type (n, 0) and where α0= 0 as an inverse image of the GKZ-system Mβ Awhere β=α:= (α2, . . . , αn)t∈Rn−1. 16 ALBERTO CASTA˜ NO DOM´ INGUEZ AND CHRISTIAN SEVENHECK 0−→ H0Rcstg†O−β S1−→ H0R◦g†O−β S1−→ H0Rg†O−β S1−→ H1Rcstg†O−β S1−→ 0. Moreover, the DV-modules HiRcstg?Oβ S1, for i∈ {−1,0,1}and ?∈ {+,†}, that appear in the above sequences are OV-free. Consequently, for any β∈Cd, calling jV∗the canonical inclusion V∗:= A1 λ0×S2,→V, we have isomorphisms of DA1 z×S2-modules FLloc S2j+ V∗R◦ cg+Oβ S1 ∼ =FLloc S2j+ V∗Rg+Oβ S1, FLloc S2j+ V∗R◦g†Oβ S1 ∼ =FLloc S2j+ V∗Rg†Oβ S1 . Proof. Following the notation of the previous definition, since Zis smooth, the excision triangle ([HTT08, Prop. 1.7.1]) corresponding to the diagram U ,→P(V∨)×V←Zgives rise to the following triangles of Radon transformations for any M ∈ Db c(DP(V∨)): RcstM −→ R◦M −→ RM −→,(8) RM −→ R◦ cM −→ RcstM −→,(9) where the second triangle is dual to the first. Note that DOβ S1 ∼ =O−1−β S1 ∼ =O−β S1. It suffices then to show the existence of the first exact sequence of the statement of the proposition and the fact that the HiRcstg?Oβ S1are constant. The existence of the exact sequences here follows by just a variation of [Rei14, Prop. 2.8], but considering the twist by y−1−βof OS1. That needs in turn [ibid., Prop. 2.5, 2.7, Lem. 2.6], but almost every argument is functorial and anyway they can be easily adapted to our context. The constancy of HiRcstg?Oβ S1can be proved as the second point of [ibid., Lem. 2.7].  Proof of Proposition 3.12. Call p1the restriction to Z∗of the projection Pn×A1 λ0×S2→Pn, abusing a bit of the notation. Consider the cartesian diagram S1×S2  k  π1//S1 g  Z∗p1 //Pn . By base change, we have that (10) k+Oβ S1×S2 ∼ =p+ 1g+Oβ S1 for any β∈Cd. Now, since p1is smooth, we have the analogous isomorphism k†Oβ S1×S2 ∼ =p+ 1g†Oβ S1 for every βas well. Now note that because of p1being non-characteristic, it is easy to show (cf. the proof of the third point of [RS17, Prop. 2.22]) that for every β (11) k†+Oβ S1×S2 ∼ =p+ 1g†+Oβ S1. Consider now the commutative diagram Z∗ p1 }} p2// jZ∗  V∗ jV∗  PnZ πZ 1 oo πZ 2 //V , where the square is also cartesian. By applying base change again, we obtain the natural transformations of functors from Db c(DPn) to Db c(DV∗) (12) p2,+p+ 1∼ =p2,+j+ Z∗πZ,+ 1∼ =j+ V∗πZ 2,+πZ,+ 1=j+ V∗R. Therefore, applying p2,+to isomorphisms (10) and (11) (and taking into account that ϕ=p2◦k, see diagram (7)) we obtain (13) ϕ+Oβ S1×S2 ∼ =p2,+k+Oβ S1×S2 ∼ =j+ V∗Rg+Oβ S1,and p2,+k†+Oβ S1×S2 ∼ =j+ V∗Rg†+Oβ S1, for any β∈Cd. IRREGULAR HODGE FILTRATION OF SOME CONFLUENT HYPERGEOMETRIC SYSTEMS 17 We need now to relate the various Radon transformations with the Fourier-Laplace transformation FL : Modh(DV∨)→Modh(DV). This is possible due to the fundamental result of d’Agnolo and Eastwood [DE03, Prop. 1]. We quote the formulation from the proof of [RS17, Lem. 2.12]. Let Bl0(V∨)⊂Pn×V∨be the blow-up of V∨at the origin and consider the commutative diagram T:= Gm×S1h// ˜ h '' πT  V∨Bl0(V∨) p oo q  V∨\ {0} j0 OO π  S1 g//Pn , where πis the canonical morphism V∨\ {0} → P(V∨), πTis the second projection, and hand ˜ hare given by (y0, y)7→ (y0, y0ya1, . . . , y0yad). Then we have the natural transformations (14) R◦ cg+∼ =FL h+π+ Tand R◦g†∼ =FL h†π+ T, of functors from Modh(DS1) to Modh(DV). Notice that although in general R◦ cand R◦are not exact, so the compositions R◦ cg+and R◦g†are precisely due to the above isomorphisms. Applying FLloc S2j+ V∗to the natural transformations in (14) we obtain the following ones of functors from Modh(DS1) to Modh(DA1 z×S2): (15) FLloc S2j+ V∗R◦ cg+∼ =FLloc S2j+ V∗FL h+π+and FLloc S2j+ V∗R◦g†∼ =FLloc S2j+ V∗FL h†π+ We are now closer to the isomorphism of the statement; let us rewrite in a different way the functors above to obtain it. Namely, FLloc S2j+ V∗FL ∼ =j+ V∗FLloc WFL ∼ =j+ V∗(jz×idW)+(jτ×idW)+FLWFL ∼ =j+ V∗(jz×idW)+(jτ×idW)+FLA1 w0. From the third of the assumptions 3.1 we know that we can decompose the morphism has h:TGm,w0×W∨A1 w0×W∨=V∨ h1jτ×idW∨, where h1is a closed embedding (cf. [RS15b, Prop. 2.1 (1), proof of Thm. 2.4, p. 213]). (Recall that w0=−τ.) It follows that (jτ×idW)+FLA1 w0h+∼ =FLA1 w0(jτ×idW∨)+h+∼ =FLA1 w0(jτ×idW∨)+(jτ×idW∨)+h1,+∼ =FLA1 w0h1,+. Summarizing, we obtain that FLloc S2j+ V∗FL h?∼ =j+ V∗(jz×idW)+FLA1 w0h1,+, where ?∈ {+,†}, because h1,+=h1,†for h1is proper. In particular, we have FLloc S2◦j+ V∗◦FL ◦h+∼ = FLloc S2◦j+ V∗◦FL ◦h†. As a consequence, we can claim using the isomorphisms in (15) that FLloc S2j+ V∗R◦ cg+∼ =FLloc S2j+ V∗R◦g†, and so, from Proposition 3.14 and applying this last natural transformation to Oβ S1, FLloc S2j+ V∗Rg†+Oβ S1 ∼ =FLloc S2j+ V∗Rg+Oβ S1. In conclusion, we finally obtain, by using the isomorphisms from (13) as well as the exactness of the functor FLloc S2, that FLloc S2H0p2,+k†+Oβ S1×S2 ∼ =FLloc S2H0ϕ+Oβ S1×S2. The last statement is an easy consequence of isomorphisms in (12) and (13), noting that FLloc S2H0p2,+k†+Oβ S1×S2 ∼ =FLloc S2H0j+ V∗Rg†+Oβ S1 ∼ =FLloc S2j+ V∗Rg+Oβ S1 ∼ =FLloc S2H0ϕ+Oβ S1×S2 is induced from the canonical morphism g†+Oβ S1,→g+Oβ S1via FLloc S2j+ V∗R. 18 ALBERTO CASTA˜ NO DOM´ INGUEZ AND CHRISTIAN SEVENHECK Consider again the space P=Pn×(A1 λ0×S2), together with the canonical closed embedding iP:Z∗,→P. We have the diagram Z∗P A1 λ0×S2 p2 iP p2, where we denote the restriction of the projection p2:P→A1 λ0×S2to Z∗by the same letter (as was done before). Since iPis proper, we have H0p2,+k†+Oβ S1×S2 ∼ =H0p2,+(iP◦k)†+Oβ S1×S2and H0ϕ+Oβ S1×S2 ∼ =H0p2,+k+Oβ S1×S2 ∼ =H0p2,+(iP◦k)+Oβ S1×S2. so that we can also write the morphism H0p2,+k†+Oβ S1×S2−→ H0ϕ+Oβ S1×S2from the proof of Proposition 3.12 as (16) H0p2,+(iP◦k)†+Oβ S1×S2−→ H0p2,+(iP◦k)+Oβ S1×S2. In order to proceed, we will have to take into account certain group actions on the spaces S1×S2, Pand A1 λ0×S2as well as some equivariance properties of the various sheaves of modules on these spaces. For the reader’s convenience, we recall some facts from [RS17, §2.4]. We consider the action of S1on S2given by µ:S1×S2−→ S2 (y1, . . . , yd, λ1, . . . , λn)7−→ (t−a1λ1, . . . , t−anλn), inducing the following three: S1×(S1×S2)−→ S1×S2 (t, (y1, . . . , yd, λ1, . . . , λn)) 7−→ (t1y1, . . . , tdyd, t−a1λ1, . . . , t−anλn), S1×P−→ P=Pn×(A1 λ0×S2) (t, ((w0:. . . :wn), λ0, λ1, . . . , λn)) 7−→ ((w0:ta1w1:. . . :tanwn), λ0, t−a1λ1, . . . , t−anλn), S1×A1 λ0×S2−→ A1 λ0×S2 (t, (λ0, λ1, . . . , λn)) 7−→ (λ0, t−a1λ1, . . . , t−anλn). It is easy to see that all these four actions are free, basically because µis free and that they all have smooth geometric quotients. These are described by the following result, which we cite from [RS17, §2.4]. For a free action of an algebraic group Gon a smooth variety Xadmitting a geometric quotient, we write X/G for such quotient. Proposition 3.15. In the above situation, put 0 S:= (Gm,t)n−d. Then the geometric quotients (S1× S2)/S1,P/S1and (A1 λ0×S2)/S1are given, respectively, by the spaces S1×0 S, 0 P:= Pn×A1 λ0×0 S, and A1 λ0×0 S. There is a canonical embedding 0 S ,→S2inducing embeddings (all denoted by ι) S1×0 S ,→S1×S2, 0 P ,→P=Pn×A1 λ0×S2, S1×A1 λ0×0 S ,→S1×A1 λ0×S2. In the sequel, we will always consider 0 Sas a subvariety of S1(as well as 0 Pas a subvariety of P etc.). There is a more direct description of the construction of 0 Sresp. of the embedding ιwhich we recall for the reader’s convenience. Namely, let Ba Gale dual of A(that is, an integer matrix IRREGULAR HODGE FILTRATION OF SOME CONFLUENT HYPERGEOMETRIC SYSTEMS 19 B∈M(n×(n−d),Z) whose columns generate kerQ(A); see Proposition 2.8). Then we have the split exact sequence of abelian groups 0−→ Zn−dB· −→ Zn−A· −→ Zd−→ 0. By applying the functor HomZ(•,Gm) to that sequence, we get the exact sequence of algebraic groups (1) −→ S1=Gd m−→ S2=Gn m−→ Gn−d m−→ (1), where the first nontrivial morphism is just µ(•,1). From this it follows that the torus Gn−d mcan be canonically identified with the geometric quotient S2/S1. Since this exact sequence of algebraic tori also splits, we can chose a section ι:0 S=S2/S1,→S2, and this choice corresponds exactly to the one in [RS17, §2.4]. We define reduced versions of the maps ϕand iP◦kby the cartesian diagrams S1×S2A1 λ0×S2 S1×0 SA1 λ0×0 S ϕ ι 0 ϕ ιand S1×S2P S1×0 S0 P iP◦k ι l ι. The mapping 0 ϕ:S1×0 S→A1 λ0×0 Sis a family of Laurent polynomials parametrized by 0 S, and we denote by 0 f:S1×0 S→A1 λ0its first component. In order to illustrate these statements, consider the main case of interest, where we have A=     1−100. . . 0 1 0 −1 0 . . . 0 . . .. . . 1 0 0 0 . . . −1     ∈M ((n−1) ×n, Z). Then d=n−1, 0 S=Gm,t and we can choose ι:0 S=Gm,t −→ S2=Gn m t7−→ (t, 1,...,1). Then we have (see also the explanation on page 7) (17) 0 ϕ:S1×Gm,t −→ A1 λ0×Gm,t (y1, . . . , yn−1, t)7−→ −1 y1−. . . −1 yn−1−t·y1·. . . ·yn−1, t Going back to the situation of a general matrix A∈M(d×n, Z) satisfying the assumptions 3.1, we state the following result, inspired from [RS17], showing that ιbehaves well with respect to all modules in question. To simplify our notation we will write M†+,S2:= H0p2,+(iP◦k)†+Oβ S1×S2and MS2:= H0ϕ+Oβ S1×S2=H0p2,+(iP◦k)+Oβ S1×S2, and analogously, M†+:= H00 p2,+l†+Oβ S1×0 Sand M:= H00 ϕ+Oβ S1×0 S=H0p2,+l+Oβ S1×0 S, where we write 0 p2:0 P=Pn×A1 λ0×0 S→A1 λ0×0 S(to distinguish this projection from the one to A1 λ0×S2considered above). Proposition 3.16. We have a morphism of DA1 λ0×0 S-modules φ:M†+−→ M inducing an isomorphism of DA1 z×0 S-modules FLloc 0 SM†+∼ =FLloc 0 SM. 20 ALBERTO CASTA˜ NO DOM´ INGUEZ AND CHRISTIAN SEVENHECK Moreover, both modules Mand M†+are non-characteristic with respect to the projection 0 p:0 P→ A1 λ0×0 Salong the subspace 0 P\0 P∗, where 0 P∗=An×(A1 λ0×0 S). Proof. It has been shown in [RS17, Prop. 2.22., Lem. 6.4.] that the morphism ι:A1 λ0×0 S ,→A1 λ0×S2 is non-characteristic for the modules M†+,S2and MS2. Consequently, we obtain the morphism of DA1 λ0×0 S-modules φ:M†+−→ M by applying ι+to the morphism M†+,S2=H0p2,+(iP◦k)†+Oβ S1×S2−→ H0ϕ+Oβ S1×S2=H0p2,+(iP◦k)+Oβ S1×S2=MS2 (see (16)). Since both the functor FLloc S2and the functor FLloc 0 Scommute with ι, we obtain the isomorphism FLloc 0 SM†+∼ =FLloc 0 SM. Finally, the fact that ιis non-characteristic for M†+,S2and MS2yields the last statement using Lemma 3.9.  Remark 3.17.From now on we will assume that the βiare real numbers, since we will use some Hodge theoretic constructions, which are not valid for arbitrary complex βi, as commented in Remark 3.6. We start by introducing certain auxiliary filtrations on the D-modules considered above which are not a priori Hodge filtrations but have an explicit description. We will see later that it is sufficient to consider these filtrations to extract the Hodge theoretic information we need. Recall that an easy calculation decomposing 0 ϕas a graph embedding followed by a projection shows that the direct image complex 0 ϕ+Oβ S1×0 Scan be represented by 0 ϕ∗Ω•+d S1×0 S/0 S[∂λ0], d −κ(β)∧ −(d0 f∧)⊗∂λ0, where 0 fis still the first component of 0 ϕ. We consider the filtration on each 0 ϕ∗Ωl+d S1×0 S/0 S[∂λ0] given by (18) Fk+l0 ϕ∗Ωl+d S1×0 S/0 S[∂λ0] = k+l X i=0 0 ϕ∗Ωl+d S1×0 S/0 S∂i λ0. One easily checks that this filtration is compatible with the differential, so that we obtain the filtered complex Fk0 ϕ∗Ω•+d S1×0 S/0 S[∂λ0], d −(κ(β)∧)−(d0 f∧)⊗∂λ0:= Fk+•0 ϕ∗Ω•+d S1×0 S/0 S[∂λ0], d −(κ(β)∧)−(d0 f∧)⊗∂λ0. Definition 3.18. We call the induced filtration F•Hi0 ϕ+Oβ S1×0 Son the cohomology sheaves Hi0 ϕ+Oβ S1×0 S the Brieskorn filtration. Notice that for i= 0, k= 0 and β= 0 the filtration step FkH00 ϕ+Oβ S1×0 S=F0Mis exactly the module M0considered in [Sab06] and was called Brieskorn lattice there because it was defined similarly to the case of isolated hypersurface singularities (see [Bri70]). On the other hand, it is well-known that l†+Oβ S1×0 Scarries a filtration FHl†+Oβ S1×0 Ssuch that l†+Oβ S1×0 S, FHunderlies a pure polarizable complex Hodge module of weight n= dim(S1) + dim(0 S), in particular, an element of MHM(0 P, C). Since 0 p2is projective, the direct image M†+= H00 p2,+l†+Oβ S1×0 Salso carries a Hodge filtration FHH00 p2,+l†+Oβ S1×0 Ssuch that M†+, FHunderlies a pure polarisable complex Hodge module (see the first point of [Sai88, Thm. 1]). We consider the shifted filtration FHsh •M†+:= FH •−dM†+. The next step is to compare this filtration to the Brieskorn filtration on Mvia the morphism φ. We will show in the next lemma that φis filtered. Notice that since the Brieskorn filtration has a priori no Hodge properties, we cannot simply deduce this result from the functorial properties of mixed Hodge modules. IRREGULAR HODGE FILTRATION OF SOME CONFLUENT HYPERGEOMETRIC SYSTEMS 21 Lemma 3.19. Consider again the morphism φ:M†+−→ Mfrom Proposition 3.16 which yields an isomorphism of DA1 z×0 S-modules after applying the functor FLloc 0 S. Then φis filtered with respect to the shifted Hodge filtration on M†+and with respect to the Brieskorn filtration on M, i.e., for every k∈Z, we have the inclusion φFHsh kM†+⊂FkM. Proof. Recall that 0 P∗=An×(A1 λ0×0 S) is the complement in 0 Pof the divisor w0= 0. Then the map lcan be decomposed into a closed embedding l1:S1×0 S ,→0 P∗and an open embedding l2:0 P∗,→0 P. Here again the fact that l1is closed follows from the third assumption in 3.1. We can further decompose l1as l1=l0◦i0 ϕ, where i0 ϕis the graph embedding of 0 ϕ. More precisely, we have the following diagram: (19) S1×0 S S1×(A1 λ0×0 S)0 P∗=An×(A1 λ0×0 S)0 P=Pn×(A1 λ0×0 S) A1 λ0×0 S 0 ϕ i0 ϕ l1 l l0 0 p20 p2 l2 0 p2 , where we denote by slight abuse of notation all projections to the last coordinates by 0 p2. Recall that M†+=H00 p2,+l†+Oβ S1×0 Sand M=H00 ϕ+Oβ S1×0 S=H00 p2,+l1,+Oβ S1×0 S. Notice that the Hodge filtration FH •M†+is induced from the filtration FH •DR•+n 0 P/A1 λ0×0 S(l†+Oβ S1×0 S) on the relative de Rham complex, where (20) FH kDR•+n 0 P/A1 λ0×0 S(l†+Oβ S1×0 S) := . . . −→ Ωn+l 0 P/A1 λ0×0 S⊗FH k+n+ll†+Oβ S1×0 S−→ . . . by the first point of [Sai88, Thm. 1]. On the other hand, we have the isomorphism of DA1 λ0×0 S-modules H00 p2,∗DRn+• 0 P∗/A1 λ0×0 Sl1,+Oβ S1×0 S∼ =M as the above diagram shows, just by the definition of H00 p2,+. The Hodge filtration FH •on the module l1,+Oβ S1×0 Scan be explicitly written down, due to the fact that l1is a closed embedding (in particular, the filtered module (l1,+Oβ S1×0 S, FH •) underlies a pure Hodge module), and then one checks that the shifted Brieskorn filtration F•[−d] := F•+don the module M(see formula (18) for the definition of the Brieskorn filtration) is induced from the filtration (21) FH kDR•+n 0 P∗/A1 λ0×0 S(l1,+Oβ S1×0 S) := . . . −→ Ωn+l 0 P∗/A1 λ0×0 S⊗FH k+n+ll1,+Oβ S1×0 S−→ . . .. Notice once again that we cannot deduce from this description that the Brieskorn filtration F•M(or its shifted version F•[−d]M) has any Hodge properties since the map 0 p2:0 P∗→A1 λ0×0 Sis not projective (and in particular the filtration from formula (18) on the direct image complex 0 ϕ+Oβ S1×0 S is not necessarily strict). We will now show that the induced morphism on global sections φ: Γ(A1 z×0 S, M†+)−→ Γ(A1 z×0 S, M) is filtered, i.e. sends the Γ(A1 z×0 S, OA1 z×0 S,)-submodule Γ(A1 z×0 S, FHsh kM†+) to Γ(A1 z×0 S, FkM). This is obviously equivalent to the statement of the proposition since A1 z×0 Sis affine. 22 ALBERTO CASTA˜ NO DOM´ INGUEZ AND CHRISTIAN SEVENHECK We have (denoting a:A1 z×0 S→ {pt}) Γ(A1 z×0 S, M†+) = a∗M†+=a∗H0R(0p2)∗DRn+• 0 P/A1 λ0×0 S(l†+Oβ S1×0 S) =H0R(a◦0p2)∗DRn+• 0 P/A1 λ0×0 S(l†+Oβ S1×0 S) = H0RΓ0 P, DRn+• 0 P/A1 λ0×0 Sl†+Oβ S1×0 S and similarly Γ(A1 z×0 S, M+) = a∗M+=a∗H0R(0p2)∗DRn+• 0 P∗/A1 λ0×0 S(l1,+Oβ S1×0 S) =H0R(a◦0p2)∗DRn+• 0 P∗/A1 λ0×0 S(l1,+Oβ S1×0 S) = H0RΓ0 P∗,DRn+• 0 P∗/A1 λ0×0 Sl1,+Oβ S1×0 S =H0Γ0 P∗,DRn+• 0 P∗/A1 λ0×0 Sl1,+Oβ S1×0 S (for the last equality, we use that 0 P∗is affine, in contrast to 0 P). Now notice that the morphism φ: Γ(A1 z×0 S, M†+)−→ Γ(A1 z×0 S, M) is induced from RΓ0 P, DRn+• 0 P/A1 λ0×0 Sl†+Oβ S1×0 S→RΓ0 P∗,DRn+• 0 P/A1 λ0×0 Sl†+Oβ S1×0 S ∼ =RΓ0 P∗,DRn+• 0 P/A1 λ0×0 Sl1,†+Oβ S1×0 S ∼ =RΓ0 P∗,DRn+• 0 P∗/A1 λ0×0 Sl1,+Oβ S1×0 S ∼ =Γ0 P∗,DRn+• 0 P∗/A1 λ0×0 Sl1,+Oβ S1×0 S, where the first morphism is simply given by restricting sections of l†+Oβ S1×0 Sfrom 0 Pto 0 P∗and the isomorphisms are due to the fact that lrestricted to 0 P∗is l1, which is proper and affine. Obviously, the restriction of the Hodge filtration on the module l†+Oβ S1×0 Sis the Hodge filtration on the module l1,†+Oβ S1×0 S=l1,+Oβ S1×0 S. Since, as has been discussed before, the filtration FH kDR•+n 0 P/A1 λ0×0 S(l†+Oβ S1×0 S) induces FH •M†+and the filtration FH kDR•+n 0 P∗/A1 λ0×0 S(l1,+Oβ S1×0 S) induces F•[−d]M, we conclude that φ: Γ(A1 z×0 S, M†+)→Γ(A1 z×0 S, M) sends Γ(A1 z×0 S, FH kM†+) to Γ(A1 z×0 S, Fk[−d]M), or, equivalently, Γ(A1 z×0 S, FHsh kM†+) to Γ(A1 z×0 S, FkM), as required.  In general, the filtrations on M†+and Mare not equal, simply because the underlying DA1 λ0×0 S- modules are not equal. They become equal after localized partial Fourier transformation as we have shown in Proposition 3.12. First we will explain that these transformations can be performed at the filtered level and how the last result can be interpreted in this context. We will use a general procedure which produces from a filtered DA1 λ0×0 S-module (N, F•) a lattice GF• 0inside FLloc 0 SN, i.e., an OA1 z×0 S-module which generates FLloc 0 SNover DA1 z×0 S. Definition 3.20. (cf. [Sab08, §1.d], [SY15, §A.1]) Let Xbe a smooth affine variety and let (N, F•) be a filtered DA1 s×X-module, which we identify with its module of global sections. Consider the algebraic microlocalization N[∂−1 s] := C[s]h∂s, ∂−1 si ⊗C[s]h∂siN. By letting act τas −∂sand ∂τas s, we consider N[∂−1 s] as a DX[τ, τ−1]h∂τi-module (which actually coincides with FLloc XN). Let now c loc be the natural localization morphism c loc : N→N[∂−1 s]. Then we define (22) GF• 0FLloc XN:= X j≥0 ∂−j sc loc(FjN), IRREGULAR HODGE FILTRATION OF SOME CONFLUENT HYPERGEOMETRIC SYSTEMS 23 notice that then GF• 0FLloc XNhas naturally the structure of a Rint A1 z×X-module. We also put for any k∈Z GF• kFLloc XN:= zk·GF• 0FLloc XN=X j≥0 ∂−(j+k) sc loc(FjN) = X j≥0 ∂−j sc loc(Fj+kN). There is an interpretation of this construction as a Fourier-Laplace transformation for Rint A1 z×A1 s×X- modules as explained in [SY15, Rem. A.3]. Using this interpretation, one can show the following fact. Lemma 3.21. Let (N, F•)be a filtered DA1 s×X-module underlying an element in MHM(A1 s×X, C)(the abelian category of complex mixed Hodge modules). Then the Rint A1 z×X-module GF• 0FLloc X(N)underlies an element of IrrMHM(X). Proof. We first define the algebraic Fourier-Laplace transformation for integrable R-modules as in [CDRS18, Defs. 3.2, 3.7]. Consider the following diagram: A1 s×XP1 s×X X j qq, where jis the canonical open embedding and qand ¯qthe respective second projections. Let As/z aff the RA1 z×As×X-module OA1 z×As×Xequipped with the z-connection zd +ds. Then for an algebraic Rint A1 z×A1 s×X-module N, we put FLR X(N) := H0q+N ⊗ As/z aff ∈Mod(Rint A1 z×X). Let Dbe the reduced divisor (P1 s\A1 s)×X={∞} × X. Then As/z ∗:= j∗As/z aff carries a natural structure of an RA1 z×P1 s×X(∗D)-module. Let Es/z ∗be the analytification of As/z ∗. As in the case of D-modules, there is the notion of strict specializability and V-filtration for Rintmodules; check [Moc15a, §2.1.2] for more details. Notice that the meromorphic function s∈ OP1 s×X(∗D) evidently extends to a map P1 s×X→P1 s whose reduced pole divisor is exactly D∼ =X, in particular, it is smooth. Then it follows from [SY15, Lem. 3.1] (using that, according to their notation, Pred =Dand e= 1) that Es/z ∗is coherent not only over RA1 z×P1 s×Xbut over V0RA1 z×P1 s×Xtoo. By [Moc15a, §2.1.2] we know that Es/z ∗is automatically specializable along Dand its corresponding V-filtration is trivial. Therefore, by construction, Es/z := Es/z ∗[∗D] = Es/z ∗(cf. [ibid., §3.1.2]). We then know by [SY15, Prop. 3.3] that Es/z underlies an algebraic, integrable pure twistor D-module Ts/z on A1 s×X. (Cf. [Sab18a, §1.6.a] for another discussion on this issue.) Now suppose that Nunderlies an algebraic, integrable mixed twistor D-module N= (N0,N, C)∈ MTMint alg(A1 s×X), then we can define its relative Fourier-Laplace transform FLMTM X(N) := H0q∗(N⊗ T s/z)∈MTMint alg(X). For notational convenience, for any R-triple K= (K0,K, C) we define the forgetful functor For(K) := K. Then we have a comparison formula, the proof of which is completely analogous to [CDRS18, Prop. 3.5]: Let Nbe an element in MTMint alg(A1 s×X), then there is an isomorphism of RA1 z×X-modules: (23) For(FLMTM X(N)) ∼ =z−1FLR X(For(N)). In other words, if Nis an RA1 z×A1 s×X-module underlying an algebraic, integrable mixed twistor D- module on A1 s×X, then the (shifted) relative algebraic Fourier-Laplace transform z−1FLR X(N) underlies an element of MTMint alg(X). 24 ALBERTO CASTA˜ NO DOM´ INGUEZ AND CHRISTIAN SEVENHECK Now let us suppose that we are given a filtered DA1 s×X-module (N, F•) underlying a complex mixed Hodge module on A1 s×X. Then its Rees module RFNis an algebraic RA1 z×A1 s×X-module and underlies an element Nin MTMint alg(A1 s×X) as well as in IrrMHM(A1 s×X). Since all the functors entering in the definition of FLMTM Xpreserve the category of irregular Hodge modules by [Sab18a, Cor. 0.5.], we know by the above comparison result (i.e. by formula (23)) that z−1FLR X(RFN) underlies an irregular Hodge module on X. Now, by [SY15, Rem. A.3] (noting that the argument there is valid for the partial transformation as well), we can identify FLR X(RFN) with zGF• 0FLloc X(N), so that GF• 0FLloc X(N)∼ =z−1FLR X(RFN). This shows that GF• 0FLloc X(N) underlies an irregular Hodge module on X, as required.  With these definitions at hand, we have the following consequence of Lemma 3.19: Corollary 3.22. In the above situation, we have GFHsh • 0FLloc 0 SM†+⊂GF• 0FLloc 0 SM Proof. This is a direct consequence of the definition in formula (22), taking into account Lemma 3.19 and the fact that the filtered morphism φinduces an isomorphism of DA1 z×0 S-modules by applying the functor FLloc 0 S. From now on, we will specify the above situation to our main example, where the matrix Ais given by A=     1−100. . . 0 1 0 −1 0 . . . 0 . . .. . . 1 0 0 0 . . . −1     . In particular, we have d=n−1, and 0 S=Gm,t. We still write M†+=H0p2,+l†+Oβ S1×Gm,t and M=H00 ϕ+Oβ S1×Gm,t . We seek to improve the inclusion of the last corollary to an equality of lattices inside the DA1 z×Gm,t -modules FLloc Gm,t (M†+)∼ =FLloc Gm,t (M). We will follow an argument from the proof of [Sab08, Lem. 4.7]. In order to do so, we have to make more explicit the structure of the module FLloc Gm,t (M)∼ =FLloc Gm,t (M†+), which is done by the next lemma. Lemma 3.23. (1) The singular locus Σ := Sing(M) = Sing(H0ϕ+Oβ S1×Gm,t )is given by Σ = [ ξ∈µn {(n·ξ·t0, t)} ⊂ A1 λ0×Gm,t, where t0is an n-th root of t, chosen without loss of generality. (2) Write D:= {0} × Gm,t ⊂A1 z×Gm,t. Then FLloc Gm,t (M)is OA1 z×Gm,t (∗D)-locally free of rank n. (3) Consider the sheaf b OA1 z×Gm,t , which is the formal completion of OA1 z×Gm,t along the divisor {0} × Gm,t. Then we have a decomposition (as sheaves on {0} × Gm,t) FLloc Gm,t (M)⊗OA1 z×Gm,t (∗D)b OA1 z×Gm,t (∗D)∼ =M ξ∈µnb Eξ⊗b Nαξ, where b Eξ:= ( b OA1 z×Gm,t (∗D), d −d(n·ξ·t0/z)) and b Nαξ= ( b OA1 z×Gm,t (∗D), d +αξdz/z), with αξ∈C. Proof. (1) It is well known that the singular locus of a Gauss-Manin system, i.e., of the topcohomology of the direct image complex ϕ+M, is nothing but the discriminant of the morphism ϕprovided that the module Mis smooth (which is the case here, since M=Oβ S1×Gm,t ). Recall (see formula (17)) that 0 ϕ(y1, . . . , yn−1, t) = 1 y1+. . . +1 yn−1+t·y1·. . . ·yn−1, t. One easily IRREGULAR HODGE FILTRATION OF SOME CONFLUENT HYPERGEOMETRIC SYSTEMS 25 checks (see, e.g., [DS04, §1.B.]) that a point (y1, . . . , yn−1)∈S1is critical if and only if y1=. . . =yn−1=: yand yn·t= 1 (and that all critical points are Morse). Then the critical values are as indicated. (2) This is a direct consequence of the second point of [DS03, Thm. 1.11], since by the discussion above the singular locus Σ satisfies the assumption (NC) of loc. cit. It is also known that the rank of FLloc Gm,t (M) equals the global Milnor number of ϕ, i.e., the numbers of critical points, which is n. (3) This follows from [Sab02, Ch. III, Thm. 5.7], since the critical values of ϕ, i.e., the eigenvalues of the pole part of z2∇z, are distinct for any t∈Gm,t.  With these preparations, we can state the next result. As has been explained at the end of section 2 from page 7 on, it is the main step to show that the Rint A1 z×Gm,t -module b Hunderlies a mixed Hodge module (that is, the content of Theorem 2.13). The explicit description of b Has a cyclic quotient by two operators can be rather easily identified with the object GF• 0FLloc Gm,t M, as we will see below in the proof of Theorem 2.13, whereas the Hodge theoretic property we want (i.e., the fact that it is an object in IrrMHM(Gm,t)) holds for GFHsh • 0FLloc Gm,t M†+. Hence we need to identify these two Rint A1 z×Gm,t -modules. Theorem 3.24. In the above situation, we have GFHsh • 0FLloc Gm,t M†+=GF• 0FLloc Gm,t M. Proof. We have already proved the inclusion GFHsh • 0FLloc Gm,t M†+⊂GF• 0FLloc Gm,t M of OA1 z×Gm,t -modules. Since both sheaves coincide outside the divisor D={0} × Gm,t, and since b OA1 z×Gm,t is O(A1 z×Gm,t,D)-flat, it is therefore sufficient to show that GFHsh • 0FLloc Gm,t M†+⊗b OA1 z×Gm,t =GF• 0FLloc Gm,t M⊗b OA1 z×Gm,t . This follows as in the proof of [Sab08, Lem. 4.7]: Using the formal decomposition result from the last Lemma, both modules can be interpreted as microlocal filtered direct images under 0 p2of two modules which coincide on 0 P∗. In these direct images, the contributions from 0 P\0 P∗vanish by the last statement of Proposition 3.16 and Lemma 3.9 (notice that inside P=Pn×A1 λ0×S2, we have 0 P∗∩ΓX∼ =S1×0 Swhere we see 0 Sas a subspace of S2via the embedding ι), and therefore both modules are equal.  We are finally able to complete the proof of Theorem 2.13. It remains to show that the Rint A1 z×Gm,t - module b H=Rint A1 z×Gm,t (z2∂z+ntz∂t−zPn i=1 αi),Qn i=1 z(t∂t−αi)−t) associated to the purely irregular hypergeometric DGm,t -module H(αi,∅) underlies an object of the category IrrMHM(Gm,t). End of the proof of theorem 2.13. Let us assume first that α1= 0, denote by αthe vector (α2, . . . , αn) and put α0:= 0. From Lemma 2.12 and Proposition 3.8 we conclude that b H∼ =ι+H0π2,∗Ω•+d S1×S2/S2[z], z (d−κ(α)∧)−df∧ ∼ =H0π2,∗Ω•+d S1×Gm,t/Gm,t [z], z (d−κ(α)∧)−d0 f∧, recall that 0 fis the first component of 0 ϕ, as written in (17). Finally, it is easy to see from Definition 3.20 that we have H0π2,∗Ω•+d S1×Gm,t/Gm,t [z], z (d−κ(α)∧)−d0 f∧∼ =GF• 0FLloc Gm,t H00 ϕ+Oα S1×Gm,t =GF• 0FLloc Gm,t M. 32 ALBERTO CASTA˜ NO DOM´ INGUEZ AND CHRISTIAN SEVENHECK Instituto de Matem´ aticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela (Spain); Fakult¨ at f¨ ur Mathematik, Technische Universit¨ at Chemnitz, 09107 Chemnitz (Germany) Fakult¨ at f¨ ur Mathematik, Technische Universit¨ at Chemnitz. 09107 Chemnitz (Germany)