Examples of hypergeometric twistor D-modules

EXAMPLES OF HYPERGEOMETRIC TWISTOR D-MODULES ALBERTO CASTA˜ NO DOM´ INGUEZ, THOMAS REICHELT, AND CHRISTIAN SEVENHECK Abstract. We show that certain one-dimensional hypergeometric differential systems underlie objects of the category of irregular mixed Hodge modules, which was recently introduced by Sabbah, and compute the irregular Hodge filtration for them. We also provide a comparison theorem between two different types of Fourier-Laplace transformation for algebraic integrable twistor D-modules. 1. Introduction In a series of papers (see [Yu14, ESY17, SY15, Sab18]), Sabbah and Yu (partly joint with Esnault) have considered a so-called irregular Hodge filtration on certain cohomology groups resp. on certain irregular D-modules. It can be seen as a generalization of the Hodge filtration on a mixed Hodge module in the sense of M. Saito. Geometrically, such a filtration arises by considering a version of the twisted de Rham cohomology of certain proper maps, and it plays (conjecturally) a role in Hodge theoretic mirror symmetry (see [KKP17]). In [Sab18], Sabbah has defined a category of irregular mixed Hodge modules, which is (up to a technical equivalence) a certain subcategory of T. Mochizuki’s category of (integrable) mixed twistor D-modules. He has proved that a rigid irreducible D-module on the projective line can be uniquely upgraded to an irregular Hodge module if and only if its formal local monodromies are unitary. Consequently, these objects come equipped with an irregular Hodge filtration and one can define irregular Hodge numbers for them. They should be seen as interesting numerical invariants attached to these differential systems, contrary to the case of arbitrary mixed twistor D-modules, where there is no obvious way to define such numbers. In [CDS17], the first and the third named author have computed that filtration and its corresponding numbers for the purely irregular hypergeometric modules, that is for systems of the form DGm/DGmP, where Pis the operator P= n Y i=1 (t∂t−αi)−t for real numbers α1, . . . , αn. Let us consider the non-commutative ring Rint Gm:= C[z, t±]hz2∂z, tz∂ti. A crucial point was to show that a certain quotient of the corresponding sheaf Rint Gmon Gmwhich restricts to the DGm,t -module DGm/DGmPon z= 1 actually underlies an object in the category IrrMHM(Gm) and the latter can be uniquely extended to an object in IrrMHM(P1). In this paper we discuss the case of more general hypergeometric D-module, that is, for quotients DGm/DGmP, where now Pis of the form P= n Y i=1 (t∂t−αi)−t m Y j=1 (t∂t−βj) for positive integers m, n and real numbers α1, . . . , αn, β1, . . . , βmsuch that there is no integer difference between any αiand βj(this is the irreducibility assumption). It is worth noticing that the presence of the factor Qm j=1(t∂t−βj) rules out the usage of the geometric arguments of [CDS17]. We obtain (see Theorem 5.7) that for certain such systems, the corresponding quotient of Rint Gmstill underlies an 2010 Mathematics Subject Classification. Primary 14F10, 32C38. Key words and phrases. D-modules, irregular Hodge filtration, twistor D-modules, Fourier-Laplace transformation, hypergeometric D-modules. The authors are partially supported by the project SISYPH: ANR-13-IS01-0001-01/02 and DFG grant SE 1114/5-1. The first named author is also partially supported by MTM2014-59456-P, the ERDF and GI-2136. The second named author is supported by a DFG Emmy-Noether-Fellowship (RE 3567/1-1). 1 2 ALBERTO CASTA˜ NO DOM´ INGUEZ, THOMAS REICHELT, AND CHRISTIAN SEVENHECK object of IrrMHM(Gm). As an application, we can completely determine the irregular Hodge filtration for all systems D/DPas above, where nis arbitrary and where m= 1. The strategy of the proof (which is rather different from that of [CDS17]) of the main theorem is to reduce these differential systems from (Fourier-Laplace transformed) A-hypergeometric D-modules (the so-called GKZ-systems of Gel’fand, Graev, Zelevinski and Kapranov, see [GGZ87], [GZK89]), but at the level of (algebraic, integrable, mixed) twistor D-modules. Notice that the paper [Moc15b] also studies twistor structures on GKZ-systems, by considering twistor D-modules associated to meromorphic functions. We use instead a central result of [RS15], where the Hodge filtration on certain GKZ-systems has been computed explicitly. Technically, the main point in our proof consists in showing that for an R-module underlying an integrable mixed twistor D-module on the affine space, the algebraic Fourier-Laplace transformation (which is defined very much the same as in the case of algebraic D-modules) coincides with the Fourier-Laplace transformation that can be defined inside the category MTM, or even IrrMHM. Along the way, we also obtain (see Theorem 4.7) that an R-module version of the GKZ-D-module underlies an irregular Hodge module provided that the parameter β∈Cdof this system satisfies a natural combinatorial condition. Notice that for the special case β= 0, this theorem can also be deduced from [Moc15b, Prop. 1.4.]. Our results give concrete representations for objects in the category MTM resp. IrrMHM, which usually are difficult to describe explicitly. We hope that a similar approach can be used to understand the irregular Hodge filtration for some higher dimensional analogues of the classical hypergeometric systems, also called Horn systems, which occur in the mirror symmetry picture for toric varieties. Acknowledgements. We would like to thank Takuro Mochizuki for communicating the proof of Lemma 2.3 to us. We would further like to thank Takuro Mochizuki and Claude Sabbah for their interest in our work and for many stimulating discussions. We are grateful to the participants of the workshop Mixed Twistor D-modules in Heidelberg in 2017 for their time and effort. We also thank the Max Planck Institute for Mathematics in the Sciences, where some part of the work presented here has been carried out. 2. Some results on Rand mixed twistor D-modules Let Xbe a complex manifold of dimension d. We denote by OXthe sheaf of holomorphic functions and DXthe sheaf of differential operators with holomorphic coefficients. Recall that DXis generated by the tangent sheaf ΘX. We put X:= A1 z×X, where the subscript means that zis the canonical coordinate on A1. Denote by pz:X→Xthe projection. We denote by RXthe sheaf of subalgebras of DXgenerated by zp∗ zΘXover OXand by Rint Xthe sheaf of subalgebras of DXgenerated by zp∗ zΘXand z2∂zover OX. In local coordinates x1, . . . , xd, they are given by OXhz∂x1, . . . , z∂xdiand OXhz2∂z, z∂x1, . . . , z∂xdi, respectively. We set Ω1 X:= z−1p∗ zΩ1 Xas a subsheaf of p∗ zΩ1 X⊗OX(∗({0} × X)), Ωp X:= VpΩ1 Xand ωX:= Ωd X. Let f:X→Ybe a morphism of complex manifolds. We consider the transfer R-modules, given by RX→Y:= OX⊗f−1OYf−1RYand RY←X:= ωX⊗RX→Y⊗f−1ωY, being respectively a (RX, f−1RY)-bimodule and a (f−1RY,RX)-bimodule. We have the inverse image and direct image functors f+(N) := RX→Y L ⊗f−1RYf−1N, f+(M) := Rf∗(RY←X L ⊗RXM), (1) between the bounded derived categories Db(RX) and Db(RY). If f:X×Y→Yis a projection and dim X=d, then f+(M) is given by f+(M) = Rf∗DRX×Y/Y(M)[d], where DRX×Y/Y(M) is the relative de Rham complex with differential d(η⊗m) = dη ⊗m+ d X i=1 dxi z∧η⊗z∂xim, EXAMPLES OF HYPERGEOMETRIC TWISTOR D-MODULES 3 the (xi)1≤i≤dbeing local coordinates on X. Let σ:Gm,z →Gm,z be the automorphism z7→ −z−1. Set S:= {z∈A1 z| |z|= 1}. If λ∈S then σ(λ) = −λ. Let E(d,d) S×X/S,c(V) be the space of C∞-sections of Ωd,d S×X/Sover any open subset Vof S×Xwith compact support and C0 c(S) the space of continous functions on Swith compact support. The space of C∞(S)-linear maps E(n,n) S×X/S,c(V)→C0 c(S) is denoted by DbS×X/S(V). This gives rise to the sheaf DbS×X/S. The abelian category R-Tri(X) consists of triples T= (M1,M2, C) where M1,M2are RX-modules and C:M1|S×X⊗σ∗M2|S×X→DbS×X/Sis a RX|S×X⊗σ∗RX|S×Xlinear morphism. If D⊂Xis a hypersurface, one similarly defines a category R-Tri(X, D) using RX(∗D) := RX⊗OXOX(∗(A1 z×D))-modules (cf. [Moc15a, §2.1] for details). Now let X:= X0×A1 tand let ΘX(log X0) be the sheaf of vector fields on Xwhich are logarithmic along X0. Let V0RXbe the sheaf of sub-algebras in RXwhich is generated by zp∗ zΘX(log X0). For z0∈A1 zwe denote by X(z0)a small neighborhood of {z0} × X. A coherent RX-module is called strictly specializable along tat z0if M|X(z0)is equipped with an increasing and exhaustive filtration V(z0) a(M|X(z0))a∈Rby coherent (V0RX)|X(z0)-modules satisfying certain conditions (cf. [Moc15a, §§ 2.1.2.1, 2.1.2.2]). This filtration is unique if it exists. Mis called strictly specializable along tif it is strictly specializable along tfor any z0. Remark 2.1.If Mis itself a coherent V0RX-module, then Mis automatically specializable along tand the corresponding filtration Va(M) exists globally and is trivial, i.e. Va(M) = Vb(M) for all a, b ∈R. If Mis a coherent RX(∗t)-module, we define similarly a filtration V(z0) a(M|X(z0)) and the notion of strict specializability along t(cf. [Moc15a, §3.1.1]). In this case we define the RX-submodules M[∗t] resp. M[!t] of M, which are locally generated by V(z0) 0Mresp. V(z0) <0M. Remark 2.2.If the coherent RX(∗t)-module Mis itself V0RXcoherent, then M[!t] = M[∗t] = M(∗t) = M. Given an RX(∗t)-triple T= (M1,M2, C) which is strictly specializable along twe can define T[!t] := (M1[∗t],M2[!t], C[!t]),T[∗t] := (M1[!t],M2[∗t], C[∗t]) (cf. [Moc15a, Prop. 3.2.1] for details). The category of filtered RX-triples (i.e. RX-triples equipped with a finite increasing filtration W) underlies the category MTM(X) of mixed twistor D-modules (cf. [Moc15a, Def. 7.2.1]). The full subcategory of objects T ∈ MTM(X) satisfying T=T[∗D] for some hypersurface D⊂Xis denoted by MTM(X, [∗D]). If Xis a smooth, algebraic variety, we denote by Xan the corresponding complex manifold. Let Xbe a smooth, complete, algebraic variety such that X ,→Xis an open immersion and D:= X\Xis a hypersurface. We can define the category of (integrable) algebraic, mixed twistor D-modules as (2) MTM(int) alg (X) := MTM(int)(Xan,[∗D]). We remark that this definition is independent of the completion up to an equivalence of categories ([Moc15a, Lem. 14.1.3]). Let f:X→Ybe a quasi-projective morphism of smooth, algebraic varieties. We take completions X⊂X, Y ⊂Yas above, such that DX:= X\Xand DY:= Y\Yand we have a projective morphism f:X→Ywhich restricts to f. For T ∈ MTMalg(X), corresponding to T ∈ MTM(X, [∗DX]), we define fi ∗T:= Hif∗T, where f∗is the direct image functor for mixed twistor D-modules arising from the one for R-modules depicted in (1). If Xis an algebraic variety, we denote by DXthe sheaf of algebraic differential operators and by RXthe sheaf of z-differential operators, where here X:= A1 z×X. We define the inverse and direct image functor in the category of algebraic RX-modules as in (1). Analogously to the construction of RX, we can consider the projection p:P1×X→X, and construct the sheaf of subalgebras of 4 ALBERTO CASTA˜ NO DOM´ INGUEZ, THOMAS REICHELT, AND CHRISTIAN SEVENHECK DP1×X(∗({∞} × X)) generated by z2∂zand zp∗ΘXover OP1×X(cf. [Moc15a, §14.4.1.1]), which will be denoted by Rint P1×X(∗∞). In that sense, an algebraic integrable RX-module gives rise to a unique Rint P1×X(∗∞)-module (cf. [ibid., Thm. 14.4.8]). The following Lemma, which will be needed later, is due to T. Mochizuki. Lemma 2.3. Given two good Rint P1×X(∗∞)-modules P1,P2and an analytic isomorphism f:Pan 1→ Pan 2, then fis induced by a unique algebraic isomorphism between P1and P2. Proof. Take a coherent OP1×X-submodule N1⊂ P1such that Rint P1×X(∗∞)⊗ N1→ P1is surjective and a coherent OP1×X-module N2⊂ P2such that both Rint P1×X(∗∞)⊗ N2→ P2is surjective and f(Nan 1)⊂ Nan 2. According to GAGA we have a morphism g:N1→ N2which after analytification is equal to the morphism Nan 1→ Nan 2induced by f. Denote by K1the kernel of Rint P1×X(∗∞)⊗N1→ P1. This gives a morphism K1→ P2which one obtains as the composition K1→ Rint P1×X(∗∞)⊗ N1 ϕ −→ Rint P1×(∗∞)⊗ N2→ P2, where ϕis induced by g. Because the induced morphism (Rint P1×X(∗∞)⊗ N1)an → Pan 2factors through Pan 1, the induced morphism Kan 1→ Pan 2is 0. Hence, we obtain that K1→ P2is 0, which means that Rint P1×X(∗∞)⊗N1→ P2factors through P1. This shows the existence. The uniqueness follows from [Ser56, Prop. 10].  Since an algebraic, integrable, mixed twistor D-module on Xgives rise to an analytic Rint P1×X(∗∞)- module which underlies an algebraic Rint P1×X(∗∞)-module by [Moc15a, Thm. 14.4.8], the Lemma above shows that we can define functors (up to canonical isomorphism) Fori: MTMint alg(X)−→ Mod(Rint X) (M1,M2, C)7→ Mifor i= 1,2, which become faithful if we impose goodness. 3. Fourier transformation of twistor D-modules In this section we define the Fourier-Laplace transformation in the categories of integrable R-modules and integrable, algebraic, mixed twistor D-modules, and we prove that these two transformations are compatible. Consider the diagram AN×b ANj// p zz q $$ PN×b PN q  ANb AN b j//b PN , where pand qare the projections to the first and second factor respectively. Consider the function ϕ=PN i=1 wi·λion AN×b AN. Let Aϕ/z aff the RA1×AN× b AN-module OA1×AN× b ANequipped with the z-connection zd+dϕ, and consider the reduced divisor D:= (PN×b PN)\(AN×b AN). Then Aϕ/z ∗:= j∗Aϕ/z aff carries a natural structure of an RA1×PN× b PN(∗D)-module. We denote by Eϕ/z ∗the analytification of Aϕ/z ∗, which is an RA1×PN× b PN(∗D)-module. Lemma 3.1. Eϕ/z ∗is strictly specializable along Dand Eϕ/z := Eϕ/z ∗[∗D] = Eϕ/z ∗. Proof. We denote the coordinates on PN×b PNby ((w0:w1:. . . :wN),(λ0:λ1:. . . :λN)), where the chart AN×b ANis embedded via the map j: (w1, . . . , wN, λ1, . . . , λN)7→ ((1 : w1, . . . , wN),(1 : λ1:. . . : λN)). By symmetry it is enough to prove the claim in the charts {w16= 0, λ06= 0},{w16= 0, λ16= 0} and {w16= 0, λ26= 0}. We will assume N≥2 and consider the chart X:= {w16= 0, λ26= 0}; EXAMPLES OF HYPERGEOMETRIC TWISTOR D-MODULES 5 the arguments with the other charts and when N= 1 go similarly. The chart Xis embedded as (x1, . . . , xN, µ1, . . . , µN)7→ ((x1: 1 : x2. . . :xN),(µ1:µ2: 1 : µ3:. . . :µN)), so that the map ϕis given on Xby 1 x1µ1(µ2+x2+Pi≥3µixi). Set DX:= A1×(D∩X) = A1× {x1·µ1= 0}. The module (Eϕ/z ∗)|Xis a cyclic RA1×X(∗DX)-module RA1×X(∗DX)/I, where the left ideal Iis generated by z∂x1+1 x2 1µ1 (µ2+x2+X i≥3 µixi), z∂x2−1 x1µ1 , z∂xj−µj x1µ1 , z∂µ1+1 x1µ2 1 (µ2+x2+X i≥3 µixi), z∂µ2−1 x1µ1 , zµj−xj x1µ1 , where j≥3. Consider the map ig:X→A1 t×Xgiven by (x1, . . . , xN, µ1, . . . , µN)7→ (x1·µ1, x1, . . . , xN, µ1, . . . , µN). The direct image ig,+(RX(∗DX)/I) is a cyclic RA1 t×X(∗(A1 t×DX))-module RA1 t×X(∗(A1 t×DX))/J0 where J0is generated by z∂x1+µ1z∂t+1 x2 1µ1 (µ2+x2+X i≥3 µixi), z∂x2−1 x1µ1 , z∂xj−µj x1µ1 , z∂µ1+x1z∂t+1 x1µ2 1 (µ2+x2+X i≥3 µixi), z∂µ2−1 x1µ1 , z∂µj−xj x1µ1 , t −x1µ1, where j≥3. Define the cyclic RA1 t×X(∗t)-module RA1 t×X(∗t)/Jwhere Jis generated by z∂x1+µ1z∂t+µ1 t2(µ2+x2+X i≥3 µixi), z∂x2−1 t, z∂xj−µj t, z∂µ1+x1z∂t+x1 t2(µ2+x2+X i≥3 µixi), z∂µ2−1 t, z∂µj−xj t, t −x1µ1, where j≥3. Then we have the following RA1 t×X-linear isomorphism RA1 t×X(∗(A1 t×DX))/J0−→ RA1 t×X(∗t)/J P·1 (x1µ1)k7→ P·1 tk. Consider the V-filtration along t= 0. The relations 1 tk= (z∂µ2)k, z∂t=−1 t z∂x1x1+1 t(µ2+x2+X i≥3 µixi) =−z∂x1x1z∂µ2−(µ2+x2+X i≥3 µixi)(z∂µ2)2 and a straightforward induction over kfor (z∂t)kshow that ig,+(RX(∗DX)/J) is a cyclic, hence also coherent, V0RA1 t×X-module. It follows from Remark 2.1 that ig,+(RX(∗DX)/J) = ig,+(RX(∗DX)/J)[∗t], and as a consequence, we are done by applying [Moc15a, §3.3.1.1] and Remark 2.2.  It follows from [SY15, Prop. 3.3] that Eϕ/z underlies an object Tϕ/z ∈MTMint alg(AN×b AN). Let us notice that the preceding lemma, as well as the similar lemma 3.6 below, are related to a more general statement in [Moc15b, Cor. 3.12] on mixed twistor D-modules associated to non-degenerate functions. However, in order to keep the paper self-contained, we prefer to give direct proofs here. We will now define a Fourier-Laplace transformation for algebraic Rint A1×AN-modules. Definition 3.2. The Fourier-Laplace transformation functor from the category of algebraic Rint A1×ANmodules to the category of algebraic Rint A1× b AN-modules is defined as c M:= FL(M) := H0q+(p+M)⊗ Aϕ/z aff , 6 ALBERTO CASTA˜ NO DOM´ INGUEZ, THOMAS REICHELT, AND CHRISTIAN SEVENHECK for any Min Mod(Rint A1×AN). Remark 3.3.Let M:= Γ(A1×AN,M) be the Rint A1×AN-module of global sections of M. The Rint A1× b ANmodule c M:= Γ(A1×b AN,c M) is isomorphic to Mas a C[z]-module and the full Rint A1× b AN-structure is given by λi·m:= −z∂wi·m, z∂λi·m:= wi·mand z2∂z·m:= z2∂z− N X i=1 z∂wiwi!·m . On the other hand, there is a similar definiton of a Fourier-Laplace transformation in the category of algebraic DAN-modules (see e.g. [Rei14, Def. 1.2]) which we also denote by FL. The Fourier-Laplace transformation for algebraic, integrable, mixed twistor D-modules is defined in the following way. Definition 3.4. The Fourier-Laplace transformation in the category of algebraic, integrable mixed twistor D-modules on ANis defined by FLMTM(M) := H0q∗(p∗M)⊗ T ϕ/z, where M ∈ MTMint alg(AN). Recall that for M= (M1,M2, C)∈MTMint alg(X) we denote by Forithe forgetful functors Fori(M) = Mifor i= 1,2. Proposition 3.5. Let M ∈ MTMint alg(AN). Then For1(FLMTM(M)) = FL(For1(M)) and For2(FLMTM(M)) = z−NFL(For2(M)). Proof. By [Moc15a, §14.3.3.3] it is clear that Forialmost commutes with p∗, more precisely we have For1(p∗(M)) = zNp+(For1(M)) and For2(p∗(M)) = p+(For2(M)). Then it is enough to prove for N ∈ MTMint alg(AN×b AN) that q+(Fori(N)⊗Aϕ/z aff )∼ =Fori(q∗(N ⊗T ϕ/z)). We have bj+q+Fori(N)⊗ Aϕ/z aff ∼ =q+j+Fori(N)⊗ Aϕ/z aff ∼ =q+j∗Fori(N)⊗ Aϕ/z aff  ∼ =Rq∗DRPN× b PNj∗Fori(N)⊗ Aϕ/z aff . Since N,Tϕ/z ∈MTMint alg(AN×b AN), there exist mixed twistor D-modules N,Tϕ/z ∈MTMint(PN× b PN,[∗D]) whose underlying R-modules are (after stupid localization along D) analytifications of the j∗Fori(N) and j∗Aϕ/z aff . Hence j∗Fori(N)⊗ Aϕ/z aff an ∼ =ForiN ⊗ T ϕ/z(∗D)∼ =ForiN ⊗ T ϕ/z, where the last equation follows from Lemma 3.1. We therefore get bj+p+Fori(N)⊗ Aϕ/z aff an ∼ =Rq∗DRan PN× b PNj∗Fori(N)⊗ Aϕ/z aff an ∼ =Rq∗DRan PN× b PNForiN ⊗ T ϕ/z ∼ =Foriq∗N ⊗ T ϕ/z. The claim follows now from Lemma 2.3, noting that the goodness is a consequence of Lemma 3.1 and [Moc15a, Thm. 14.4.15].  EXAMPLES OF HYPERGEOMETRIC TWISTOR D-MODULES 7 We have the following variant, which will be used in the next section. Consider the diagram AN×Gm j// p zz q $$ PN×P1 q  ANGm b j//P1 and let ψ:= w1·t+w2+. . . +wN. Similarly as above we define the RA1×AN×Gm-module Aψ/z aff , being OA1×AN×Gmendowed with the zconnection zd+dψ. As in the other case, we can consider the divisor H:= (PN×P1)\(AN×Gm) and obtain the RA1×PN×P1(∗H)-module Aψ/z ∗:= j∗Aψ/z aff . In the same vein as before, we will denote by Eψ/z ∗the RA1×PN×P1(∗H)-module being the analytification of Aψ/z ∗. The following Lemma is similar to Lemma 3.1. Lemma 3.6. Eψ/z ∗is strictly specializable along Hand Eψ/z := Eψ/z ∗[∗H] = Eψ/z ∗. Proof. We denote the coordinates on PN×P1by ((w0:w1:. . . :wN),(u:t)), where the chart AN×Gmis embedded via the map j: (w1, . . . , wN, t)7→ ((1 : w1:. . . :wN),(1 : t)). We will assume N≥3 and consider the chart X:= {w26= 0, u 6= 0}; the other charts behave similarly, as well as the case N= 1,2. The chart Xis embedded as (x1, . . . , xN, u)7→ ((x1:x2: 1 : x3:. . . :xN),(u: 1)). On this chart the map ψis given by 1 x1(x2 u+1+x3+. . .+xN). Set HX:= A1 s×(H∩X) = A1 s×{x1·u= 0}. The module (Eψ/z ∗)|Xis a cyclic RX(∗HX)-module RX(∗HX)/I, where the left ideal Iis generated by z∂x1+1 x2 1x2 u+1+x3+. . . +xN, z∂x2−1 x1u, z∂xj−1 x1 , z∂u+x2 x1u2, with j≥3. Consider the map ig:X→A1 s×Xgiven by (x1, . . . , xN, u)7→ (x1·u, x1, . . . , xN, u). Analogously as in Lemma 3.1, the direct image ig,+(RX(∗HX)/J) is a cyclic RA1 s×X(∗(A1 s×HX))- module RA1 s×X(∗(A1 s×HX))/J0where J0is the left ideal generated by z∂x1+uz∂s+1 x2 1x2 u+1+x3+. . . +xN, z∂x2−1 x1u, z∂xj−1 x1 , z∂u+x1z∂s+x2 x1u2, s −x1u, and j≥3. Define the cyclic RA1 s×X(∗s)-module RA1 s×X(∗s)/Jwhere Jis generated by z∂x1+uz∂s+1 s2x2u+u2+x3u2+. . . +xNu2, z∂x2−1 s, z∂xj−u s, z∂u+x1z∂s+x1x2 s2, s −x1u, where j≥3. We have the following RA1 s×X-linear isomorphism RA1 s×X(∗(A1 s×HX))/J0−→ RA1 s×X(∗s)/J P1 (x1u)k7→ P1 sk. Consider the V-filtration along s= 0. The relations 1 sk= (z∂x2)k, z∂s=−1 sz+uz∂u+x2 s=−z·z∂x2−uz∂uz∂x2−x2(z∂x2)2 and a straightforward induction over kfor (z∂s)kshow that ig,+(RX(∗DX)/J) is a coherent V0RA1 t×Xmodule. As in the previous lemma, this shows the claim.  8 ALBERTO CASTA˜ NO DOM´ INGUEZ, THOMAS REICHELT, AND CHRISTIAN SEVENHECK It follows again from [SY15, Prop. 3.3] that Eψ/z underlies an object Tψ/z ∈MTMint alg(An×Gm). Definition 3.7. (1) The Fourier-Laplace transformation with respect to the kernel ψin the category of algebraic RA1×AN-modules is defined as FLψ(M) := H0q+(p+M)⊗ Aψ/z aff , for any M ∈ Mod(RAN). (2) Analogously, the Fourier-Laplace transformation with respect to the kernel ψin the category of algebraic, integrable twistor D-modules on ANis defined by FLψ MTM(M) := H0q∗(p∗M)⊗ T ψ/z, for any M ∈ MTMint alg(AN). We get the following result for the kernel ψ. Proposition 3.8. Let M ∈ MTMint alg(AN). Then For1(FLψ MTM(M)) = z1−NFLψ(For1(M)) and For2(FLψ MTM(M)) = z−NFLψ(For2(M)) . Proof. We have, by [Moc15a, §14.3.3.3], For1(p∗(M)) = zp+(For1(M)) and For2(p∗(M)) = p+(For2(M)). The rest of the proof carries over almost word for word from Proposition 3.5, using Lemma 3.6.  4. GKZ systems and irregular Hodge modules Let A= (aki) be a d×Ninteger matrix with columns (a1, . . . , aN). We define NA:= N X i=1 Nai⊂Zd and similarly for ZAand R≥0A. Throughout this section we assume ZA=Zdand NA=Zd∩R≥0A . Set AN:= Spec (C[w1, . . . , wN]) and b AN:= Spec (C[λ1, . . . , λN]) and LA:= (`= (`1, . . . , `N)∈ZN: N X i=1 `iai). Definition 4.1. The GKZ-hypergeometric system Mβ Ais the cyclic Db AN-module Db AN/I, where Iis the left ideal generated by Ek:= N X i=1 akiλi∂λi−βk,for k= 1, . . . , d, and `:= Y `i>0 ∂`i λi−Y `i<0 ∂−`i λi,for l∈LA. The GKZ-hypergeometric system Mβ Ais the Fourier-Laplace transform of the cyclic DAN-module ˇ Mβ A:= DAN/J, where Jis the left ideal generated by ˇ Ek:= N X i=1 aki∂wiwi+βk,for k= 1, . . . , d, and ˇ `:= Y `i>0 w`i i−Y `i<0 w−`i i,for l∈LA. EXAMPLES OF HYPERGEOMETRIC TWISTOR D-MODULES 9 The semigroup ring C[NA]⊂C[t± 1, . . . , t± d] is naturally a C[w1, . . . , wN]-module under the isomorphism C[w1, . . . , wN]/((ˇ `)`∈LA)−→ C[NA] wi7−→ tai, where we are using the multi-index notation tai:= Qd k=1 taki k. We set SA:= C[NA]. Notice that the rings C[w1, . . . , wN] and SAcarry a natural Zd-grading given by deg(wi) = ai. This is compatible with the grading on the Weyl algebra DAN:= Γ(AN,DAN) given by deg(wi) = aiand deg(∂wi) = −ai. Definition 4.2. ([MMW05, Def. 5.2]) Let Pbe a finitely generated Zd-graded C[w1, . . . , wN]-module. An element α∈Zdis called a true degree of Pif the graded part Pαis non-zero. A vector α∈Cdis called a quasi-degree of Pif αlies in the complex Zariski closure qdeg(P) of the true degrees of Pvia the natural embedding Zd,→Cd. Consider the set of strongly resonant parameters of A: sRes(A) := N [ j=1 sResj(A), where sResj(A) := {β∈Cd|β∈ −(N+ 1)aj+qdeg(SA/(taj))}. Consider as well the torus Gd m:= Spec (C[t± 1, . . . , t± d]), together with the torus embedding h:Gd m−→ AN (t1, . . . , td)7→ (ta1, . . . , taN). The following proposition is a slight generalization of the results of Schulze and Walther [SW09, Thm. 3.6, Cor. 3.8]. Proposition 4.3. ([RS15, Prop. 2.11]) Let Abe a d×Ninteger matrix satisfying ZA=Zdand NA=Zd∩R≥0A. Assume that β6∈ sRes(A). Then H0h+Oβ Gd m∼ =ˇ Mβ A, where Oβ Gd m ∼ =DGd m/DGd m·(∂t1t1+β1, . . . , ∂tdtd+βd) For β∈Rd, the D-module Oβ Gd munderlies the complex mixed Hodge module pCH,β Gd m. Hence for β∈Rd\sRes(A) the D-module ˇ Mβ Aunderlies the complex mixed Hodge module H0h∗pCH,β Gd m. The Hodge filtration on ˇ Mβ Acan be explicitly computed, provided that βbelongs to a certain set AAof so-called admissible parameters β. We recall its definition from [RS15, p. 11]: Let c:= a1+. . . +aN and define for all facets Fof R≥0Athe uniquely determined primitive, inward-pointing, normal vector nFof F, such that hnF, Fi= 0 and hnF,NAi ⊂ Z≥0. Set eF:= hnF, ci ∈ Z>0. The set of admissible parameters of Ais then defined by AA:= \ Ffacet {R·F−[0,1/eF)·c}. Theorem 4.4. ([RS15, Thm. 3.16]) For β∈AAthe Hodge filtration on ˇ Mβ Ais equal to the order filtration shifted by N−d, i.e. FH p+N−dˇ Mβ A=Ford pˇ Mβ A. Let us define the cyclic RA1×AN-module ˇ Nβ A:= RA1×AN/Jz, where Jzis the left ideal generated by ˇ Ez k= N X i=1 akiz∂wiwi+zβk,for k= 1, . . . , d, and ˇ `=Y `i>0 w`i i−Y `i<0 w−`i i,for l∈LA. 16 ALBERTO CASTA˜ NO DOM´ INGUEZ, THOMAS REICHELT, AND CHRISTIAN SEVENHECK But His irreducible, so its only endomorphism is the identity and then the twistor D-module underlying His unique. 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 the assumption on the αi and the βj. Then it gives rise to a unique pure integrable twistor D-module b Hpr on P1by [Moc11, Thm. 1.4.4] and [Sab18, Rem. 1.40]. In addition, its underlying DP1-module Hpr is rigid, as Hwas. As a consequence, we can invoke [ibid., 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]. Then we must have, as was just shown, b H0∼ =b H, so that the extension b Hpr of b His unique, and we are done.  Remark 5.8.Let us consider the last theorem for the case m=n, that is, the case of regular hypergeometric systems. Consider b Has a RA1 z×Gm-module only, as such it is isomorphic to RA1 z×Gm/(H), where now H=Qm i=1 z(t∂t−αi)−tQm j=1 z(t∂t−βj). RA1 z×Gmis graded by degree in z(where zhas degree 1), and since His homogenous (which is not the case if n6=m), we see that b His a graded RA1 z×Gm-module. It is obviously strict, i.e. it has no z-torsion, and then by [SS18, A.2.5(5)], we see that b His the Rees module of a filtered DGm-module, namely, the (regular) hypergeometric module H(αi;βj) together with the filtration by order of differential operators. Notice also that if n=m, we have P=z2∂z+εz, which implies that b Hhas an action by z∂zand that if we write b H=⊕kb Hk (grading with respect to z), then for any m∈b Hk, we have (z∂z)(m)=(k−)m. Now suppose that we have n=mand that additionally the hypotheses of the last theorem are satisfied, then since b H(αi;βj) is the unique object in IrrMHM(Gm) (lying actually in the essential image of MHM(Gm)) with underlying DGm-module H(αi;βj), it is the Rees module of the filtered module (H(αi;βj), FH •), where FH •denotes the Hodge filtration of the complex variation of Hodge structures on H(αi;βj). Hence FH •H(αi;βj) = Ford •H(αi;βj) in this case. Moreover, if we put Rk:= k Y i=1 (t∂t−αi) for k= 0, . . . , n −1 (where R0:= 1), then (Rk)k=0,...,n−1is an OGm-basis of H(αi, βj) and yields a splitting of the Hodge filtration FH •. In particular, we obtain that the Hodge numbers hp(H(αi;βj)) = dim FH k/FH k−1are all equal to one. This is consistent with [Fed18, Thm. 1] (up to an overall shift, as noticed in that theorem) in the version of [CDS17, Prop. 2.6], since under the assumption of Theorem 5.7, the function #{j:βj< αk}is constant. We will finish this section with a calculation of an irregular Hodge filtration, similar to the last section of [CDS17]. In that reference, the authors computed such a filtration in the case where the hypergeometric D-module had a purely irregular singularity at infinity, that is, it was of type (n, 0). It is immediate to see that for modules of type (n, 1), the second assumption of Theorem 5.7 holds true, so that we obtain an explicit description of the Rint A1 z×Gm-module underlying the irregular Hodge module with associated DGm-module H(α1, . . . , αn;β). In the sequel, we are going to compute the irregular Hodge filtration of such modules of type (n, 1). Let us recall the conventions and notations used in [CDS17, §4] (cf. [Sab18, Not. 2.1]). We will deal with the classical hypergeometric D-module H=H(αi;β), where the αiand βare n+ 1 real numbers belonging to the interval [0,1). We will denote by b Hboth its associated algebraic, integrable twistor D-module on Gmand its underlying Rint A1 z×Gm-module (as in the statement of Theorem 5.7). From now on, we will write X,θXand τXmeaning the products A1 z×Gm,t,X × Gm,θ, and X × A1 τ, respectively, where θ= 1/τ. Finally, we will write τX0=X × {τ= 0} ⊂ τX. Theorem 5.9. Let real numbers α1, . . . , αn, β ∈[0,1) be given. Suppose that α1≤. . . ≤αnand that moreover αi−β /∈Zfor all i= 1, . . . , n. For each k= 1, . . . , n, set ρ(k) = −(n−1)αk+k. Then the jumping numbers of the irregular Hodge filtration of H=H(αi;β)are, up to an overall real shift, the numbers ρ(k). The irregular Hodge numbers are the multiplicities of those jumping numbers, or equivalently, the nonzero values of |ρ−1(x)|, for xreal. EXAMPLES OF HYPERGEOMETRIC TWISTOR D-MODULES 17 Moreover, let να(k) = d−α+k−ε−(n−1)αk+1e(recall from Theorem 5.7 that ε=β−Pn i=1 αi+n). Let us consider the operators ¯ Qk= (−(n−1))k k Y i=1 (t∂t−αi) for k= 0, . . . , n −2(where the empty product equals one) and ¯ Qn−1= (−(n−1))n−1 n−1 Y i=1 (t∂t−αi) + (−(n−1))n−1t(β−α1) 1 + α1−αn ¯ Q0. Then, the irregular Hodge filtration Firr •His given by Firr α+jH=M k:j≥να(k) OX¯ Qk. Remark 5.10.In general, the procedure given below can be of use to find an explicit expression for the irregular Hodge filtration, not only the numbers, of any hypergeometric of type (n, m), provided both assumptions from Theorem 5.7 are fulfilled. However, the calculations become soon too cumbersome to be included here. Proof. We will mimic the arguments of [CDS17, §4], providing almost no proof of the claims which are similar to some therein. We must first consider the rescaling of b H: this is the inverse image θb H:= µ∗H(as OθX-module), endowed with a natural action of Rint θXas depicted in [Sab18, 2.4] (note that θ=τ−1), where µis the morphism given in [ibid., Not. 2.1] by µ:θX → X (z, t, θ)7→ (zθ, t). In this sense, we can apply the same argument of [CDS17, Prop. 4.1] to get that the Rint θX-module θb Hassociated with b Hcan be presented as Rint θX/(P, θR, θH), where P=z2∂z+ (n−m)tz∂t+εz as in Theorem 5.7, θR=z2∂z−zθ∂θand θH= n Y i=1 zθ(t∂t−αi)−tzθ(t∂t−β). Now we have to invert θto obtain an Rint τX(∗τX0)-module τb H, to work in the setting given by [Sab18, §2.3]. In this sense, we will denote by τb Hthe Rint τX(∗τX0)-module (idX×(j◦inv))∗θb H, where inv : Gm,θ →Gm,τ is the inversion operator θ7→ τ−1and j:Gm,τ ,→A1 τis the canonical inclusion. Then it is easy to see that τb H=Rint τX(∗τX0)/(P, τR, τH), with Pas always, τR=z2∂z+zτ∂τand τH= n Y i=1 z τ(t∂t−αi)−tz τ(t∂t−β). The next step is forming the basis of τb Has a OτX(∗τX0)-module. Let it be given by Qk= (−(n−1))k k Y i=1 z τ(t∂t−αi) for i= 0, . . . , n −2 and Qn−1= (−(n−1))n−1 n−1 Y i=1 z τ(t∂t−αi) + (−(n−1))n−1t(β−α1) 1 + α1−αn Q0. It is indeed a basis: we can use the expressions of τRand Pto replace the classes of zτ∂τand z2∂z, respectively, in terms of zt∂t. Now τb His generated as a OτX(∗τX0)-module by the powers of zt∂t, and we can get rid of those of exponent greater than n−1 using τH. The remaining npowers can be expressed as a linear combination of the Qi, forming a triangular matrix (almost diagonal in fact), so the latter conform a basis as well. 18 ALBERTO CASTA˜ NO DOM´ INGUEZ, THOMAS REICHELT, AND CHRISTIAN SEVENHECK One could wonder about the odd expression of the Qi. In the case with no betas of [CDS17], the basis considered there was formed just by the successive products Qk i=1 z τ(t∂t−αi), up to some constant. In this case, such a basis does not provide a connection matrix solving the Birkhoff problem with a diagonal matrix as a coefficient of the pole at infinity in z, which would give us a way to read the spectrum from that matrix (cf. [GMS09, Prop. 4.8]). As a consequence, we have to adapt such initial basis, and that is how we get the Qi. Let us write the connection matrix explicitly. Let c= (β−α1)/(1 + α1+αn), in such a way that Qn−1= (−(n−1))n−1 n−1 Y i=1 z τ(t∂t−αi)+(−(n−1))n−1ctQ0. A similar (but longer) calculation to the proof of [CDS17, Lem. 4.3] shows that the integrable connection arising from the Rint τX(∗τX0)-module structure associated with τb Hhas the following matrix form: ∇Q=Q(τA0+zA∞)dz z2+−τA0+zA0 ∞dt (n−1)zt −(τA0+zA∞)dτ zτ . There, if n > 2, A0,A0 ∞and A∞are the matrices (5) A0=      0· · · −(−(n−1))n−1ct 0 1...(−(n−1))n−1(c+ 1)t ...0. . . 1 0      , A0 ∞= diag((n−1)α1,...,(n−1)αn) and A∞= diag(0,1, . . . , n −1) −εIn−A0 ∞. If n= 2, we have (6) A0=ct c(c+ 1)t2 1 (c+ 1)t, A0 ∞=α10 0α2and A∞= diag(0,1) −εI2−A0 ∞. Finally, the irregular Hodge filtration is obtained from a suitable V-filtration along the divisor τ= 0 defined on τb H, which is called τV-filtration (the new symbol τVis to make clear the variety over which we are working; note the same convention in [Sab18], from Remark 2.20 on). We are actually to define a filtration on τb H, and then prove that it equals the τV-filtration, following [Moc15a, §2.1.2]. Let us consider then τUατb H:= (n−1 X k=0 fkτνkQk:fk∈ OτX, max(k−(n−1)αk+1 −ε−νk)≤α), τU<ατb H:= (n−1 X k=0 fkτνkQk:fk∈ OτX, max(k−(n−1)αk+1 −ε−νk)< α), (7) for any α∈R. The τUατb Hform an increasing filtration, indexed by the real numbers but with a discrete set of jumping numbers, such that ττUατb H=τUα−1τb Hfor any α(those are conditions i and ii’ in [Moc15a, §2.1.2]). As usual, the graded piece associated with αis GrτU ατb H=τUατb H/τU<ατb H. In (7), all the exponents νkof the powers of τaccompanying the fkQksatisfy that νk≥ −α+ k−(n−1)αk+1 −ε. Then we can define the steps of the filtration in the same alternative way as in [CDS17, Rem. 4.5] as the free OτX-modules of finite rank (8) τUατb H= n−1 M k=0 OτX·τνα(k)Qk, where να(k) = d−α+k−ε−(n−1)αk+1e. With that expression, it is clear that the graded pieces GrτU ατb Hare GrτU ατb H= n−1 M k=0 OX·τνα(k)Qk, EXAMPLES OF HYPERGEOMETRIC TWISTOR D-MODULES 19 which are strict RX-modules (condition iv in [Moc15a, §2.1.2]). The next step in the proof is proving that τb His strictly R-specializable along τX0and its τV-filtration is actually given by the τUατb H. Although the proof is similar to that of [CDS17, Prop. 4.6], we have to adapt it a bit to our case here. After what we already showed, it remains to show conditions iii’ and v of [Moc15a, §2.1.2] and prove that the τUατb Hare coherent V0RX-modules. Let us start by the second condition. Consider then the mappings p,egiven by (p,e) : R×C−→ R×C (β, ω)7−→ (β+ 2<(z¯ω),−βz +ω−¯ωz2). We must check that the operator zτ∂τ−e(β, ω) is nilpotent on the graded pieces GrτU ατb Honly for a finite amount of (β, ω)∈ K := {β+ 2<(z0¯ω) = α}, for any value z0of z. Moreover, those (β, ω) should belong in fact to R× {0}(cf. [Sab18, §1.3.a]), if we want to obtain the R-specializability. Take then (β, ω)∈ K and fτνQk∈τUατb H, with f∈ OτX. We must have that k−(n−1)αk+1−ε−ν≤α. Assume that n > 2 and k < n −2. Thanks to the matrix form (5) we know that (zτ∂τ−e(β, ω))fτνQk=zτ∂τ+ (ν+ (n−1)αk+1 +ε−k+β)z−ω+ ¯ωz2(f)τνQk−fτν+1Qk+1. Recall that the αiare increasingly ordered, lying within the interval [0,1). Thus fτν+1Qk+1 lives in τUατb H, for k+ 1 −(n−1)αk+2 −ε−ν−1≤((k+ 1) −(n−1)αk+2 −ε)−(k−nαk+1 −ε)) −1 + α≤α. Now we should look at what happens to the class of fτν+1Qk+1 in the α-graded piece of τb H. Note that [fτνQk]6= 0 if and only if ν+ (n−1)αk+1 +ε−k+α= 0, so (zτ∂τ−e(β, ω))fτνQk=zτ∂τ+ (β−α)z−ω+ ¯ωz2(f)τνQk−fτν+1Qk+1 = =zτ∂τ−2<(z0¯ω)z−ω+ ¯ωz2(f)τνQk−fτν+1Qk+1. Now notice that τdivides τ∂τ(f), so in fact zτ∂τ(f)τνQk∈τUα−1τb Hand then we can further reduce our expression to (zτ∂τ−e(β, ω))fτνQk= (−ω−2<(z0¯ω)z+ ¯ωz2)fτνQk−fτν+1Qk+1. On the other hand, τν+1Qk+1 does not vanish either in GrτU ατb Hif and only if αk+2 =αk+1. Indeed, we know that ν+(n−1)αk+1 +ε−k+α= 0, so doing the same as before, k+1−(n−1)αk+2 −ε−ν−1 = α+ (n−1)(αk+2 −αk+1) and the claim follows. Furthermore, in order to (zτ∂τ−e(β, ω)) to vanish, we should impose that ω= 0, just by looking at the coefficients of the powers of zin the expression for f. If k=n−2, we obtain from (5) that (zτ∂τ−e(β, ω))fτνQn−2=zτ∂τ+ (ν+ (n−1)αn−1+ε−(n−2) + β)z−ω+ ¯ωz2(f)τνQn−2 −fτν+1Qn−1+fτν+1(−(n−1))n−1ctQ0. Since −(n−1)α1−ε−ν−1≤ −(n−1)(α1−αn−1+ 1) + α < α because αn−1< α1+ 1, the last summand above belongs to τU<ατb H, and then the argument can follow as with k < n −2. Now if k=n−1, then everything would be the same again as before except we get the additional summand −τν+1Qk+1, which becomes −fτν+1(−(n−1))n−1(c+ 1)tQ1, whose class vanishes in the graded piece under consideration, too. Indeed, 1−(n−1)α2−ε−ν−1≤ −(n−1)(α2−αn+ 1) + α < α, for αn< α2+ 1. In conclusion, (zτ∂τ−e(β, ω))lfτνQkcan only vanish in GrτU ατb Hif α=β(and then ω= 0), and does not do so until we get to an index k+lsuch that αk+lis strictly bigger than αk. Since there is a finite set of indexes, (zτ∂τ−e(β, ω)) is nilpotent, of nilpotency index nat most. 20 ALBERTO CASTA˜ NO DOM´ INGUEZ, THOMAS REICHELT, AND CHRISTIAN SEVENHECK When n= 2, we notice from (6) that we have two possibilities. If k= 0, everything is the same as with k=n−2 for n > 2, and if k= 1, (zτ∂τ−e(β, ω))fτνQ1=zτ∂τ+ (ν+α2+ε−1 + β)z−ω+ ¯ωz2(f)τνQ1 +fτν+1(c+ 1)tQ1+fτν+1c(c+ 1)t2Q0. Here the argument runs similarly as in the general case. Condition iii’ can be rephrased as zτ∂ττUατb H ⊆ τUατb H, using that τUατb H=ττUα+1τb H, and that follows essentially from the same argument used to prove condition v above. Last, since V0RX= OτXhz∂t, zτ∂τi, it is clear from the computations above and the alternative expression (8) for the filtration steps that they are cyclic V0RX-modules, and then coherent. Summing up and noting that all the calculations performed were in fact independent of z0,τb His strictly R-specializable along τX0 and the τU•τb Hform its τV-filtration. We can finally show the expression for the irregular Hodge filtration and then the irregular Hodge numbers like in [CDS17, Thm. 4.7]. Since we know that b Hunderlies an object in IrrMHM(Gm,t) by Theorem 5.7, we deduce by [Sab18, Def. 2.52] that b His well-rescalable (cf. [ibid., Def. 2.19]) and so we can apply [ibid., Def. 2.22]. After formula (8), we clearly have i∗ τ=zτVατb H=τVατb H/(τ−z)τVατb H=M k OXzνα(k)¯ Qk, which is free z-graded of finite rank. Denote by πthe projection X → Gm,t. Then, the z-adic filtration on π∗H[z−1] induces a filtration on i∗ τ=zτVατb H, given by Fri∗ τ=zτVατb H:= M s≤r M k:να(k)≤s OGm,t ¯ Qk zs. Then, GrFi∗ τ=zτUατb His the Rees module associated to a new good filtration Firr α+•Hon H, for some k= 0, . . . , n −1, which is the irregular Hodge filtration. More concretely, Firr •His given by Firr α+jH=M k:να(k)≤j OGm,t ¯ Qk. Therefore, its jumping numbers are −ε+j−1−(n−1)αjfor j= 1, . . . , n. Since the irregular Hodge filtration is defined up to an overall real shift, we can normalize the jumping numbers to j−(n−1)αj and the irregular Hodge numbers will be their multiplicities.  References [CDS17] Alberto Casta˜no Dom´ınguez and Christian Sevenheck, Irregular Hodge filtration of some confluent hypergeometric systems, preprint arXiv:1707.03259 [math.AG], 2017. [ESY17] H´el`ene Esnault, Claude Sabbah, and Jeng-Daw Yu, E1-degeneration of the irregular Hodge filtration, J. Reine Angew. Math. 729 (2017), 171–227, with an appendix by Morihiko Saito. [Fed18] Roman Fedorov, Variations of Hodge structures for hypergeometric differential operators and parabolic Higgs bundles, Int. Math. Res. Not. IMRN (2018), no. 18, 5583–5608. [GGZ87] I. M. Gel’fand, M. I. Graev, and A. V. Zelevinski˘ı, Holonomic systems of equations and series of hypergeometric type, Dokl. Akad. Nauk SSSR 295 (1987), no. 1, 14–19. [GMS09] Ignacio de Gregorio, David Mond, and Christian Sevenheck, Linear free divisors and Frobenius manifolds, Compositio Mathematica 145 (2009), no. 5, 1305–1350. [GZK89] I. M. Gel’fand, A. V. Zelevinski˘ı, and M. M. Kapranov, Hypergeometric functions and toric varieties, Funktsional. Anal. i Prilozhen. 23 (1989), no. 2, 12–26. [Kat90] Nicholas M. Katz, Exponential sums and differential equations, Annals of Mathematics Studies, vol. 124, Princeton University Press, Princeton, NJ, 1990. [KKP17] Ludmil Katzarkov, Maxim Kontsevich, and Tony Pantev, Bogomolov-Tian-Todorov theorems for LandauGinzburg models, J. Differential Geom. 105 (2017), no. 1, 55–117. [MMW05] Laura Felicia Matusevich, Ezra Miller, and Uli Walther, Homological methods for hypergeometric families, J. Amer. Math. Soc. 18 (2005), no. 4, 919–941. [Moc11] Takuro Mochizuki, Wild harmonic bundles and wild pure twistor D-modules, Ast´erisque (2011), no. 340, x+607. [Moc15a] , Mixed twistor D-modules, Lecture Notes in Mathematics, vol. 2125, Springer, Cham, 2015. EXAMPLES OF HYPERGEOMETRIC TWISTOR D-MODULES 21 [Moc15b] , Twistor property of GKZ-hypergeometric systems, preprint arXiv:1501.04146, 2015. [Rei14] Thomas Reichelt, Laurent Polynomials, GKZ-hypergeometric Systems and Mixed Hodge Modules, Compositio Mathematica (150) (2014), 911–941. [RS15] Thomas Reichelt and Christian Sevenheck, Hypergeometric Hodge modules, preprint 1503.01004 [math.AG], 2015. [Sab18] Claude Sabbah, Irregular Hodge theory (with the collaboration of Jeng-Daw Yu), M´em. Soc. Math. Fr. (N.S.) (2018), no. 156, vi+126. [Ser56] Jean-Pierre Serre, G´eom´etrie alg´ebrique et g´eom´etrie analytique, Ann. Inst. Fourier, Grenoble 6(1955–1956), 1–42. [SS18] Claude Sabbah and Christian Schnell, The MHM Project,http://www.cmls.polytechnique.fr/perso/ sabbah/MHMProject/mhm.html, 2018. [SW09] Mathias Schulze and Uli Walther, Hypergeometric D-modules and twisted Gauß-Manin systems, J. Algebra 322 (2009), no. 9, 3392–3409. [SY15] Claude Sabbah and Jeng-Daw Yu, On the irregular Hodge filtration of exponentially twisted mixed Hodge modules, Forum Math. Sigma 3(2015), e9, 71. [Yu14] Jeng-Daw Yu, Irregular Hodge filtration on twisted de Rham cohomology, Manuscripta Math. 144 (2014), no. 1-2, 99–133. E-mail address:[email protected] E-mail address:[email protected] E-mail address:[email protected] 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) Mathematisches Institut, Universit¨ at Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg (Germany) Fakult¨ at f¨ ur Mathematik, Technische Universit¨ at Chemnitz, 09107 Chemnitz (Germany)