Algebraic aspects of hypergeometric differential equations

We review some classical and modern aspects of hypergeometric differential equations, including $A$-hypergeometric systems of Gel'fand, Graev, Kapranov and Zelevinsky. Some recent advances in this theory, such as Euler-Koszul homology, rank jump phenomena, irregularity questions and Hodge theoretic aspects are discussed with more details. We also give some applications of the theory of hypergeometric systems to toric mirror symmetry.


Introduction
Notational conventions. We use Italic letters M for rings, variables and modules; calligraphic letters D for sheaves; Roman letters FL for functors; Gothic letters for prime ideals p and points x of spaces. Lattice elements a are in Roman bold; coordinate sets t and other sets of functions or operators ∂ in Italic bold. The continuing interest in hypergeometric functions stems to some extent from the fact that they are often solutions to very appealing linear differential equations taken from physics. For example, the Bessel functions J ±r (x) of the first kind arise as solutions to a linear second order equation that shows up in heat and electromagnetic propagation in a cylinder, vibrations of circular membranes, and more generally when solving the Helmholtz or Laplace equation. Indeed, such connections to physics through differential equations prompted the first studies of (specific) hypergeometric functions. However, hypergeometric functions also appear in many other parts of mathematics: as we will see soon, each time an action of an algebraic torus on a space is observed, one can expect to find some differential equation of hypergeometric type connected to this situation. The abundance of toric varieties in geometry explains why there are so many different interesting hypergeometric functions. We discuss in Section 5 below one prominent case where hypergeometric differential equations prove to be useful: the so-called mirror symmetry phenomenon for certain smooth toric varieties. Other recent applications that are beyond the scope of this article include the holonomic gradient method in algebraic statistics ( [HNT17]) or Feynman integral computations in quantum field theory ( [Nas16], [Kla19], [de 19], [FCCZ20]).
As it turns out, it is exactly the type of differential equation satisfied by a function that determines whether the function should be considered as hypergeometric, since these force the right kind of recursions on the series. The most successful approach to generalize hypergeometric differential equations to several variables was initiated by Gel ′ fand, Graev, Kapranov and Zelevinsky in the 1980s, and some of the features of this theory form the topic of this article. We start with some motivating examples. While this integral cannot be solved in closed form, it can be developed into a convergent Taylor series where a i = 1/(2i + 1), so that is hypergeometric. ♦ The univariate hypergeometric functions are classified by the rational function a i+1 /a i . More precisely, suppose that a i+1 /a i = P (i)/Q(i) where P, Q ∈ C[i] are monic with P = p j=1 (i + α j ) and Q = q j=1 (i + β j ). Then the univariate hypergeometric function associated to P, Q is p F q (α 1 , . . . , α p ; β 1 , . . . , where a 0 = 1 and a i+1 a i = (i + α 1 )(i + α 2 ) · · · (i + α p ) (i + β 1 )(i + β 2 ) · · · (i + β q ) .
Example 1.2 (The error function, part II). It follows from (1) that erf(z) is, up to the factor 2z/ √ π, equal to 1 F 1 (1/2; 3/2; −z 2 ), where 1 F 1 (1/2; 3/2; z) = 1 + z 3 + z 2 10 + z 3 42 + z 4 216 + z 5 1320 + · · · is the Kummer confluent function which encodes all intrinsic analytic and combinatorial properties of erf(x) and, with θ z = z d dz , satisfies the differential equation θ z (θ z + 1/2) • (f ) − z(θ z − 1/2) • (f ) = 0. (3) The particular shape of this equation will be used in the next section for a conversion process from univariate hypergeometric functions to A-hypergeometric ones. ♦ In the following example we document how hypergeometric functions arise naturally from differential forms with parameters. The computation was apparently already known to Kummer; compare [BK86] for details. In modern terms, it represents the birth of the notion of a variation of Hodge structures.
defines for each z ∈ C {0, 1} a smooth curve E z over C. Its projective closure E z ⊆ P 2 C meets the line at infinity in a single point and is smooth as long as z ∈ {0, 1, ∞}. The natural projection from E z to C via "forgetting v" is generically 2 : 1 and branches at 0, 1, z; the induced map E z −→ P 1 C also branches at infinity. The differential 1-form ω z := du/v is everywhere holomorphic and nowhere zero on E z ; the existence of this "form of the first kind" in Riemann's language makes the elliptic curve E z a Calabi-Yau manifold in modern terms. The "form of the second kind" ω ′ z := ω z /(u − z) has a unique pole, at u = z, at which it is residuefree. Considering v = v(u, z) as dependent variable and writing ω z , ω ′ z in terms of u and z, one notes that ∂ ∂z (ω z ) = 1 2 ω ′ z , and (compare especially [BK86,Page 685 the differential on the right being taken in u, v with z constant (and noting that on E one has d(u(u − 1)(u − z)) = 2v dv). Let λ ∈ H 1 (E z ; Z) ≃ Z ⊕ Z and set I 1 (λ) = λ ω z and I 2 (λ) = λ ω ′ z , multivalued functions on E z defined via elliptic integrals. The differential equations for ω z , ω ′ z imply (compare [BK86,Lemma 12]) that I 1 (λ) and I 2 (λ) are solutions to (4) f ′′ − qf ′ = pf, with singularities at 0, 1 and ∞. It is the special case 1 = 2a = 2b = c of the general Gauß hypergeometric differential equation with solution space basis given by Gauß' hypergeometric functions which have singularities at 0, ∞ and 1, ∞ respectively. Suppose λ z , λ ′ z are the standard basis (the minimal geodesics) for the first homology group of the torus E z . Then two elementary (but non-trivial) computations reveal: (1) analytic continuation of the solution space basis F = (F 1 , F 2 ) T around the points z = 0 and z = 1 corresponds to multiplication of F by M 0 = 1 0 −2 1 and M 1 = 1 2 0 1 respectively; (2) the map π : P 2 C {(1, 0, 0), (1, 1, 0), (0, 0, 1)} −→ P 1 C , wu(u − w) ← z 0 , wv 2 − u 3 − u 2 w ← z 1 , is a bundle with fiber E z1/z0 that admits an Ehresmann connection. In particular, the cohomology classes of the fibers allow parallel transport. The induced vector bundle with fiber H 1 (E z ; Z) = Zλ z + Zλ ′ z admits a monodromy action, lifting the loops around z = (0, 1) and z = (1, 1). Analysis of the geometry of π shows that this monodromy is given again by the actions of M 1 and M 2 respectively. More abstractly, the D-module on the base of π corresponding to the derived direct image of the structure sheaf on the source of π, also known as the Gauß-Manin system, has monodromy action via M 1 , M 2 .
On the complement of the points 0, 1, ∞ this D z -module is a vector bundle with a flat connection. The fibers of this vector bundle are the cohomology groups H 1 (E z1/z0 ; C). This vector bundle is actually a variation of pure Hodge structures of weight 1 where the (1, 0)-part is generated by the differential form ω z , the variation of this (1, 0)-subbundle being described by (4).
It follows that, up to scalars, I 1 (λ z ) = F 1 (z), I 2 (λ z ) = F 2 (z). In particular, the ratio τ (z) = I 1 (λ z )/I 2 (λ z ) is the modulus of the elliptic curve in the sense that the fiber over z is isomorphic to the quotient of C by Z + √ −1τ · Z. We will take up the discussion of Hodge structures associated to more general univariate hypergeometric operators (see equation (7) below) later in Section 4 (see page 33). ♦ 1.2. From univariate to GKZ and back. In the 1980s, the Russian school around I.M. Gel ′ fand found a universal way of encoding univariate hypergeometric functions by way of certain systems of PDEs that arise from an integer matrix A and complex parameter vector β. We start with the general definition and then explain how univariate hypergeometric functions arise as solutions of these D-modules.
Notation 1.4. In the first three sections of this article, A = (a 1 , . . . , a n ) ∈ Z d×n denotes an integer matrix with d rows and n columns. In the last two sections, A will still be integer, but at least sometimes of size (d + 1) × (n + 1). ♦ For convenience, we place the following constraints on the matrix A; they make concise statements possible, or at least easier to make. Na j ⊆ ZA inside Z d . Throughout we assume that • the group ZA generated by A agrees with Z d (A is full ); • the semigroup NA contains no units besides 0 (A is pointed ). We note that pointedness of A is equivalent to the existence of a group homomorphism from Z d to Z that is positive on every a j .

♦
We now give the definition of the main character of our story.
Definition 1.6 (A-hypergeometric system, [GGZ87]). Fix A ∈ Z d×n as in Convention 1.5 and choose β ∈ C d . Let be the n-th Weyl algebra over C. Here x = x 1 , . . . , x n , ∂ = ∂ 1 , . . . , ∂ n , and ∂ j is identified with the partial differentiation operator ∂ ∂xj . We also let denote the polynomial subring.
Letting θ j stand for x j ∂ j , the Euler operator E i is For each u ∈ Z n in the kernel of A its box operator is Finally, the hypergeometric ideal and module to A, β are

♦
Before we embark on a general discussion of these modules we wish to distinguish two special subclasses that will play a lead role.
Definition 1.7. The matrix A is homogeneous if the following equivalent properties are satisfied: • there is a group homomorphism from Z d to Z that sends every a j to 1 ∈ Z; • the vector (1, 1, . . . , 1) is in the row span of A; • the ideal I A is standard graded and thus defines a projective variety inside projective (n − 1)-space.
depending on a complex parameter vector β ∈ C d , It is not hard to verify that a hypergeometric function defined by the integral (5) is annihilated by both the Euler operators and the box operators [GKZ90,Ado94] but it took a decade to arrive at the general formulation given here.
It turns out that every univariate hypergeometric function arises as a solution of an A-hypergeometric system; we sketch next the steps to construct the proper A, β. The general hypergeometric univariate differential equation is It is elementary, but not always trivial, to bring a differential equation derived from a series expansion of a hypergeometric function into this shape; it may require changes of variables in z. Note that p F q (α; β; z) is a solution to the special form as one can see from applying the two operators to the power series (2).
Let v and c be the vectors with entries v j and c j respectively. For 2 F 1 (equal to the function F 1 in Example 1.3), v = (1, 1, −1, −1) while for the Kummer confluent function 1 F 1 , v = (1, 1, −1). Now, in order to manufacture A and β from equation (6), choose an integral matrix A such that Z · v = ker A and set β = A · c. Then the solutions of H A (β) (in other words, the functions annihilated by every operator in this left ideal) "contain the solutions to (6)" in the following sense.
Example 1.9 (The GKZ-system to the Kummer confluent function). Consider the system of partial differential equations Equation (8) forces any solution u to be homogeneous (and of degree −1/2) under the grading that attaches the weights (1, 0, 1) to (x 1 , x 2 , x 3 ). Similarly, Equation (9) asserts that u is homogeneous of weight zero if (x 1 , x 2 , x 3 ) → (0, 1, 1). It follows that one can write is of bi-degree (−1/2, 0), and g is a univariate function. Set z = x 1 x 2 /x 3 and write Enforcing the vanishing of ∂ 1 ∂ 2 − ∂ 3 on u(x 1 , x 2 , x 3 ) as suggested by Equation (10) implies the recurrence relations for all i, and the starting condition is of bi-degree (−1/2, 0), we infer b = −c = 1/2 and thus the recurrence is showing that g(z) essentially agrees with the Kummer confluent function. ♦ Example 1.10 (GKZ-system to 2 F 1 ). Take the equation (7) with p = q = 2 and c = (1, c, a, b). Then v = (1, 1, −1, −1) and the matrix A can be chosen as annihilate each solution, so every monomial x u in the power series expansion of every solution to the A-hypergeometric system must satisfy the three conditions For a monomial x u , we call A · u ∈ ZA the A-degree of x u . Then, every solution u(x 1 , x 2 , x 3 , x 4 ) can be written as a univariate function g in x1x4 x2x3 , multiplied by a monomial of A-degree β. As in the previous example, one can use the fact that v kills u to show that g satisfies the Gauß hypergeometric differential equation. ♦ Of course, the kernel of A being Z · v means that A ∈ Z (n−1)×n and I A = ( v ) is principal. On the other hand, the A-hypergeometric paradigm also encodes multivariate hypergeometric series of higher rank (namely n − d) when d < n − 1. The solutions to H A (β) use n variables and satisfy d homogeneities, so that effectively they are functions in n − d independent quantities. Some aspects of the translation between the two setups is discussed in [BMW19b]. The advantage of the A-hypergeometric point of view is that it allows hypergeometric functions to be studied with methods coming from algebraic geometry, commutative algebra, and the theory of torus actions. We describe in the following sections some of the advances and some of the new problems that have been created through these new techniques.
1.3. Solutions. While we do not focus very much on solutions of A-hypergeometric systems in this survey, it is only fair to indicate to some extent the development of the understanding of their solution space over time. We also refer the reader to Remark 3.14 below, where we list and discuss some more references, after having explained issues like irregularity and slopes of hypergeometric systems.
Classically, functions were considered as hypergeometric if they could be developed into a hypergeometric series. They typically arose from specific differential equations and the hypergeometricity was a consequence of the recurrence relations that came out of the differential equation. While introducing A-hypergeometric systems, Gel ′ fand and his collaborators Graev, Kapranov and Zelevinsky developed a similar paradigm for the multi-variable homogeneous case, see Definition 1.7. With setup as in Section 2, so A · γ = β and L A the kernel of A, the series formally is a solution of H A (β). Assuming a certain amount of genericity for γ (such as non-resonance, see Definition 2.7) the article [GZK89] also finds that the regions of convergence of these series contain an open cone of the same shape as R ≥0 .
The series approach to solving differential equations of hypergeometric type was then taken further by Sturmfels, Saito and Takayama in their book [SST00] through the technique of Gröbner bases. As part of this mechanism, triangulations arise. The connection between certain special solution series on one side and and triangulations on the other appears already in [GZK89]. In the homogeneous normal case (see Definition 1.7) it can be used to count the number of solutions as the simplicial volume of the convex hull of the columns of A; [SST00] provides various generalizations.
The first functions that were identified as hypergeometric were the Γ-type integrals t a (1 − t) b (1 − zt) c dt of Euler for the Gauß hypergeometric function. In [GKZ90], the authors consider integrals  [Sch73] of the Euler-Gauß hypergeometric differential equations whose solution space is spanned by algebraic functions. The case of all p F p−1 was dealt with much later by Beukers and Heckman in [BH89] as part of their study of the monodromy. For irreducible such equations with real parameters α 1 , . . . , α p , β 1 , . . . , β p−1 set β p = 1. Their exponentials on the unit circle are interlaced provided that the images of α i and β j are encountered alternatingly on a trip around the unit circle. Then [BH89] shows that interlacing is equivalent to the solution space of the differential equation being spanned by algebraic functions. Other cases were characterized in [Sas77,BCW92] For saturated irreducible homogeneous A-hypergeometric systems M A (β) with rational β, Beukers discovered the following fact about the number of algebraic solutions. Let C A,β = (β + ZA) ∩ (R ≥0 A) and consider it as a module over the semigroup NA. Let σ A (β) be the number of generators of C A,β over NA. Then, Beukers shows in [Beu10] that σ A (β) never exceeds the volume of A, and equality of σ A (kβ) = vol(A) for all 1 ≤ k ≤ D coprime to the least common denominator D of β 1 , . . . , β d happens precisely when the solution space is spanned by algebraic functions. We remark that irreducibility is linked to non-resonance (compare Definition 2.7) by [Beu11, Sai11, SW12].
The story for inhomogeneous (i.e., confluent) systems is more complicated, both theoretically and algorithmically. Since the solutions do not need to lie in the Nilsson ring, a systematic search in the sense of [SST00] using Gröbner bases is not possible. Nonetheless, in [ET15] an idea of Adolphson [Ado94] is completed that casts solutions of non-resonant A-hypergeometric systems as integrals Here, γ is a continuous family of real d-dimensional topological cycles in the torus, on which the integrand decays rapidly at infinity in the sense of Hien [Hie09]. This was also already studied in the context of integrals from hyperplane arrangements by [KHT92].

Torus action and Euler-Koszul complex
In this section we start exploring algebraic properties of the system H A (β) by introducing a homological tool from [MMW05] that has proved to be very successful: the Euler-Koszul complex. It has been used to study the number of solutions, their monodromy, and several other aspects. We refer to the start of Subsection 1.2 for basic notations and assumptions regarding A.
2.1. Torus action and A-grading. Given a D A -module Q, its Fourier transform Q is equal to Q as a C-vector space and carries a D A := C[ξ] ∂ structure given by for any m ∈ Q.
This action induces a grading we refer to this as the A-grading. There is a natural extension to D A if one sets deg(x j ) = −a j that makes every Euler operator A-graded of degree zero. The coordinate ring of the orbit closure through (1, . . . , 1) is the toric ring Remark 2.
The following sets are then in one-to-one correspondence: Toric category and Euler-Koszul technology. The following set of constructions and results is taken from [MMW05].
Note that E i − β i ∈ D A can be viewed as a left D-linear endomorphism on A-graded D A -modules M by sending a ZA-homogeneous y ∈ M to and that these morphisms commute with one another. The properties of the Euler-Koszul complex are most pleasant when N is in the category of toric modules. These are A-graded R A -modules that have a finite composition series whose successive quotients are ZA-shifted quotients of S A .
Remark 2.4. There is a generalization in [SW09] to quasi-toric (i.e., certain non-Noetherian A-graded) modules that is useful for the interplay of Euler-Koszul complexes on local cohomology modules or on localizations such as C[ZA]. ♦ By [MMW05], short exact sequences 0 −→ N ′ −→ N −→ N ′′ −→ 0 of toric modules give rise to long exact sequences of Euler-Koszul homology modules that are all holonomic (see Definition 2.12). Moreover, vanishing of H A,0 (N ; β) implies vanishing of all H A,i (N ; β) and this vanishing is equivalent to −β not being in the quasi-degrees of N .
Remark 2.5. Euler-Koszul complexes were initially defined for the study of the size of the solution space of A-hypergeometric systems [MMW05], but have turned out to be remarkably successful when investigating other issues such as irregularity (see section 3 and [SW08] an open resp. closed embedding. We set We denote the Fourier transform of M A (β) by M A (β) and its corresponding quasi-coherent sheaves by M A (β) and M A (β) respectively. Using the definition of the Fourier transform one easily sees that M A (β) has support on the toric variety X A . In [SW09] the parameters β were identified for which there is an isomorphism T between the Fourier transform of M A (β) and the direct image under h A of the twisted structure sheaf The relevant definition is the following one.  Theorem 2.10. Let A ∈ Z d×n be as above, then the following statements are equivalent Remark 2.11. The idea of linking M A (β) to the direct image (h A ) + O β T originates with [GGZ87] where it was shown that β non-resonant gives the desired isomorphism. The precise computation in Theorem 2.10 comes from [SW09]. These results were refined and extended to the strongly resonant case in [Ste19a,Ste19b] where Steiner uses a combination of direct and proper direct image functors. ♦ 2.4. Holonomicity, Rank, and Singular Locus. Suppose M = D A /I is some left D A -module, and M = D C n /I the associated sheaf of D C n -modules. Then its analytification M an = D an C n /D an C n I is obtained by replacing D C n by the sheaf D an C n of analytic linear differential operators on C n where now I ⊂ D C n ⊂ D an C n generates a left ideal of analytic linear differential operators. Choose x ∈ C n and denote stalks by subscripts. Consider the functor from germs of left D an A,x -modules to vector spaces (Notice that we do not consider derived solutions here, the use of the symbol Sol differs from many other texts on Dmodules). If M an = D an C n /D an C n I then η ∈ Sol x (M an ) corresponds to the analytic solution η(1 + D an C n I ) near x. The dimension of the vector space of solutions to M at x is the rank of M at x. When we mean the rank at a generic point x we speak of just the rank of M .
Typically, Sol x (M an ) is infinitely generated. But for the select class of holonomic modules it is always finite.
Definition 2.12. Any principal D A -module (resp. D an C n -module) M (resp. M ) with generator m has a natural order filtration F ord is generated by the cosets of ∂ u with |u| ≤ k. The notion readily extends to any module with chosen set of generators and is well-behaved under analytification.
If M = D an C n is the sheaf of differential operators itself, the associated graded object is on the stalk isomorphic to the regular ring O x [y] where y = y 1 , . . . , y n is the set of symbols to ∂ 1 , . . . , ∂ n . For any M (resp. M ), the associated graded object gr F (−) becomes a module over gr F (D A ) (resp. gr F (D an C n )). The module is holonomic if the associated graded module has Krull dimension n. ♦ It was shown in [GGZ87, GZK89] that many, and then in [Ado94] that in fact all A-hypergeometric systems are holonomic. This was extended in [MMW05,SW09] to all Euler-Koszul homology modules derived from quasi-toric input.
By [SKK73,Gab81], the characteristic variety is always involutive and has all components of dimension n or larger. This implies that holonomic modules have finite length and satisfy a Krull-Remak-Schmidt theorem (have well-defined sets of simple composition factors with multiplicity taken into account). Moreover For many important A-hypergeometric systems, a search of explicit natural power series solutions leads to rank many independent solutions, compare [GGZ87,SST00]. It was claimed in [GZK89] that the rank of M A (β) is where vol(A) is the (simplicial) volume of A, a purely combinatorial quantity given by the quotient of the measure of the convex hull of the origin and the columns of A, divided by the measure of the standard n-simplex. Adolphson [Ado94] pointed at a possible flaw in the argument, and [ST98] eventually provided a counter-example that is worth looking at.
The toric ideal I A is homogeneous here, defining the pinched rational normal space curve. In [SST00] it is shown that series solution methods based on weight vectors and the computation of certain initial ideals of H A (β) always lead to volume many independent series solutions, as long as A is homogeneous. This generalized the naïve series written out in [GGZ87,GZK89] to the case where logarithmic terms can appear in the series solutions.
For almost all β, the rank of M A (β) in a generic point is 4, spanned by functions where the dots indicate a (usually infinite) series of terms ordered by the weight vector (0, 1, 2, 0). (The particular weight is immaterial, but it needs to be sufficiently generic; this one is so for this example). If one now deforms β into (1, 2) then the four independent solutions above degenerate into a linearly dependent set of rank three. On the other hand, the functions 2)). It follows that the "rank jumps at β = (1, 2)", from 4 to 5 = 4 − 1 + 2. ♦ Shortly after the discovery of rank jumps, the case of homogeneous monomial curves was completely discussed in [CDD99]: the "holes" of NA (the finitely many elements of (R ≥0 A ∩ ZA) NA) are exactly the rank-jumping parameters, and each rank jump is by 1. It was then shown in [MMW05] that as β varies, the rank of M A (β) is upper-semicontinuous, so that it can only go up under specialization (formation of a limit) of β. In fact, [MMW05,Cor. 9.3] shows that the exceptional set E A of points where rank exceeds volume is Zariski closed and equals a certain subspace arrangement. To understand the origins of E A one must view the local cohomology modules H i ∂ (S A ) with i < d as quasi-toric modules; their elements are then witnesses to the failure of S A to be Cohen-Macaulay, while the union of their quasi-degrees forms the exceptional arrangement. The fact, also observed in [MMW05], that this arrangement has codimension at least two explains why finding rank-jumps at all turned out to be very hard and involved extensive computer experiments in [ST98].
Example 2.14 (Continuation of Example 2.13). In Example 2.13, d = 2 and so E A can be at most a finite set of isolated points. The local cohomology H 0 Each degree component in S A and its monomial localizations are 1dimensional C-spaces; we use this to depict these localizations in theČech complex by dots as follows: In this picture, the blue area indicates the directions in which the semigroup in question extends, black dots are the elements of A and the red dot indicates a "missing" element in the semigroup. Taking cohomology "dot-by-dot" one identifies the local cohomology groups H 1 m (S A ), H 0 m (S A ) as claimed. It is remarkable that the components of the H 1 m (S A )-cocycle are precisely the "new" solutions that appear at β = (1, 2) that do not deform to other β. While this is not always literally true, a weaker form is typical and an explanation of this phenomenon involving Laurent polynomials is given in [BFM18,BZFM16a], especially for d = 2. Compare also Remark 3.14. ♦ Remark 2.15. In [Ber11] it is proved that there is a purely combinatorial recipe (involving the relative positioning of β to the degrees of NA) that determines the rank of M A (β). The procedure to arrive at the exact rank is very involved. The only known closed rank formula is for non-jumping parameters, where the rank is just the volume. The best known general bound is exponential [SST00], in the sense that the rank of M A (β) is bounded above by 2 2d vol(A). sharp. It was shown in [MW07] that rank jump examples of the form rk(M A (β)) = vol(A)+d−1, for any d. This is improved in [FF13] to the existence of a ∈ R greater than 1 and families of matrices A (d) of size d × n d and with parameters β (d) such that the rank of M A (d) (β (d) exceeds a d vol (A). It would be interesting to know how far the bound from [SST00] is from the the worst examples that exist. ♦ There is an open subset of C n on which the solutions for M A (β) form a vector bundle of rank rk(M A (β)). The complement (the singular locus of the module) of this set is algebraic, cut out by the A-discriminant, a product of individual discriminants to polynomial systems, one for each face of the cone over A. For a very detailed discussion on this, see the books [GKZ94], and [SST00]. If one moves from general to special x, rank can go down due to singularities in the solutions. In contrast to rank in generic points, rank at special x is not known to be uppersemicontinuous. For the case of A as in Example 2.13, this is worked out in [Wal18], which discusses the more general question of stratifying C n by the restriction diagrams, which encode the behavior of the D-module theoretic (derived) pull-back to x ∈ C n ; the elementary pull-back just counts rank at x.

2.5.
Better behaved systems and contiguity. For each β ′ = a j + β there is a natural contiguity morphism of degree a j , induced by right multiplication with ∂ j on S A through the Euler-Koszul functor. The existence of these morphisms is a consequence of the fact that Since elements in I A act as zero on S A , any composition of contiguity morphisms of fixed total degree γ ∈ NA acts the same way as morphism Contiguity morphisms have turned out to be a very useful tool in the study of A-hypergeometric systems since for k ≫ 0, c β+kaj ,β+(k+1)aj and c β−(k+1)aj ,β−kaj are isomorphisms (and one can determine explicit bounds in terms of A, β for k being sufficiently big). Contiguity maps have been used in [Sai01] to identify combinatorially the isomorphism classes of A-hypergeometric systems, in [Wal07] to study irreducibility and holonomic duality of M A (β) as D A -module, and in [Rei14,RS20] for investigating the Hodge module structure on certain M A (β). For a study of Gauß hypergeometric functions via contiguity operators see [Beu07].
On the level of solutions, a map in the reverse direction is induced that literally takes the derivative by x j . For certain applications in mirror symmetry it is desirable to know that every contiguity operator induces an isomorphism on (the solutions of) M A (β). In case one has a generic β, this is automatic. But in practical situations it is more likely that β is integer, or at least resonant. In the present context, resonance encapsulates the lack of genericity of a parameter β to admit contiguity isomorphisms (in both directions). Resonance and contiguity operators were refined and used in [Ado94, Sai01, Sai11, Oku06, CDRV11, SW12, Beu11, Beu16] to study reducibility and general structure of M A (β). Now consider the quasi-toric module F A equal to the ring C[ZA]. It arises as the localization of S A at all ∂ j , or alternatively at one monomial whose degree is in the interior of R ≥0 A. By definition, multiplication by ∂ j on F A is an isomorphism, and therefore the same applies to the generalized A-hypergeometric system that arises as the Euler-Koszul homology H A,0 (F A ; β), for every β. Since F A is a maximal Cohen-Macaulay S A -module, there is no other Euler-Koszul homology, [MMW05,SW09].
This module H A,0 (F A ; β) was studied in [BPH13,BH06] and termed better behaved GKZ-system. A variant of these systems, considered in [Moc15b], can be described as the Euler-Koszul homology of the normalization of S A , i.e. as H A,0 (C[R ≥0 A∩Z d ]; β). We will make below in section 4 some comments on how the Hodge theoretic considerations described there relies to the main result of [Moc15b].

Irregularity
In this section we discuss regularity issues of hypergeometric D-modules; this is a multi-variate form of essential singularities. We start with discussing more general filtrations than the one by order. A combinatorial object can be derived from this process that governs the convergence behavior of solutions to A-hypergeometric systems near coordinate hyperplanes. Via results of Laurent and Mebkhout we discuss a generalized classical Fuchs criterion this gives information on the irregular solutions.

The Fuchs criterion and regularity.
A univariate function f (t), analytic on a small open disk around t = 0 but singular at t = 0, can behave in two essentially different ways: the growth of f (t) as t → 0 could be bounded by a polynomial, or not. In the former case, f has a pole, in the latter an essential singularity. If f arises as solution to a differential equation we say 0 is a regular singular point of the equation in the first, and an irregular singular point in the second case.
For linear differential equations P • f (z) = 0 in the local parameter z, Fuchs gave the following practical procedure for determining regularity of the origin. If O 0 := C{z} is the ring of convergent power series near z = 0, write P as a linear combination m being the order of P , and p k = ∞ i=n k c k,i z i ∈ O 0 with c k,n k = 0 indicating the lowest order term of p k (z). Writing ∂ z for differentiation by z, for a monomial z r ∂ s z we use the two weights Then plot for each k the weights of c k,n k ∂ k z in the (F, V )-plane: The shaded region (the Fuchs polygon of the operator) is the lower left convex hull of the (finitely many) points so obtained. It is, by definition, stable under shifts in negative F -and V -direction, and hence unchanged under analytic automorphisms that keep the origin fixed (this is a consequence of taking the lower left hull).
Two cases arise, indicated in the picture: (1) The Fuchs polygon has one vertex, in the upper right corner (left).
(2) There are two or more corners. This is tantamount to the boundary of the shaded region having one or more finite boundary segments with slopes different from 0 and −∞ (right). Fuchs' criterion (see [Gra84,Inc44] for a detailed account) states that P has a regular singularity at the origin if and only if the Fuchs polygon of P has no slopes.
Regular differential equations are much better behaved than irregular ones, both theoretically and practically. On the theoretic side, they form an ingredient of the Riemann-Hilbert correspondence that links regular holonomic D-modules to perverse sheaves, which for irreducible modules restricts to a bijection with intersection cohomology complexes; on the practical side regular differential equations are amenable to the Frobenius method since their solutions come from the Nilsson ring [Kas84,Meb80,Meb84,SST00].
In higher dimensions, the concept of regularity is more difficult. One way of defining it proceeds via pullbacks: the D-module M on the analytic space C n is regular if and only if the pullback of M along any analytic morphism ι : ∆ * −→ C n , where ∆ * is a punctured disk, leads to a module with regular singularities at the origin on ∆ * . The problem is that there are many such morphisms to be tested.
Laurent [Lau87] and later with Mebkhout [LM99] found a way to translate regularity in more than one variable into a condition that resembles the Fuchs criterion. For that, we need to discuss filtrations and initial ideals on D-modules in more detail.
3.2. Initial ideals and triangulations. A general technique to understand (noncommutative) algebraic structures is the reduction to a simpler (commutative) situation by applying a grading with respect to a filtration. For D-modules, the filtration by the order of differential operators leads to the characteristic variety which carries various bits of information on the D-module. The process of grading is rather cumbersome but can be performed algorithmically in various situations using Gröbner basis methods. The simplest case is that of a generic weight vector because the resulting graded ideal will be monomial. The content of this subsection is based on [SST00] and [Stu96].
So, let L = (L 1 , . . . , L n ) ∈ Q n be a generic weight vector on R A ; genericity is needed to assure that gr L (I A ) is a monomial ideal. (Over R there are weights L that are generic for all ideals of R A simultaneously. There is no rational weight with this property, but for a finite number of ideals a Zariski open set of the weight space consists of generic weights.) Example 3.1. For the matrix A = 1 0 1 0 1 1 , with columns indicated with bullets, the following picture sketches the possible initial ideals that arise from the weights in the family L t = 1 1 t , t > 0. Plotted left, with hollow bullets, are the points a j /L t j .
PSfrag replacements Collinearity of all three plotted points equates with L-homogeneity of I A . ♦ Definition 3.2. Associated to the generic weight L and the R A -ideal I is an initial simplicial complex Σ L I that arises as follows. A collection τ of indices contained in [n] forms a face of Σ L I if and only if there is no monomial in gr L (I) whose support is precisely τ . Put another way, Σ L I is the simplicial complex whose Stanley-Reisner ideal is the radical of gr L (I). If For example, suppose I A is the principal ideal generated by ∂ 1 ∂ 2 ∂ 3 − ∂ 4 ∂ 2 5 . Then I A admits two distinct monomial initial ideals whose corresponding simplicial complexes are: The generic weight L also induces a triangulation of [n] as follows. Consider the pointsÂ = {(a j , L j ) ∈ R d × R} 1≤j≤n . The faces of the triangulation are those faces of the cone R ≥0Â ofÂ that are visible from the point (0, −∞); these are exactly those faces whose outer normal vectors have negative last component. A triangulation of [n] is regular (or coherent ) if it arises this way for some L. This property is strongly tied to A, and not all triangulations of A have to be regular. The collection of regular triangulations of A turns out to be in (the obvious) bijection with the initial complexes of A. There is a third combinatorial object associated to L and A, namely the collection S (gr L (I A )) of standard pairs of gr L (I A ). A standard pair (∂ b , σ) of the monomial ideal I is a monomial and a subset of [n] such that For example, if the monomial ideal is (∂ 4 ∂ 2 5 ) the standard pairs are (1, {1, 2, 3, 4}), (∂ 5 , {1, 2, 3, 4}), and (1, {1, 2, 3, 5}). The standard pairs yield immediately a decomposition into irreducible ideals by For I as above we obtain I = (∂ 5 ) ∩ (∂ 2 5 ) ∩ (∂ 1 4 ). The standard pairs hence contain all information needed to recover I and its triangulations. In particular, the facets of Σ L A are precisely the subsets σ that are listed in the standard pairs. Example 3.3. We consider Example 3.1 from this new angle. We fix the weights L 1 = L 2 = 1 and vary the weight t = L 3 . For L 3 < 2, gr L I A = ∂ 1 ∂ 2 and the facets of Σ L A are {1, 3}, {2, 3}. We could interpret this as the complex of faces, not containing 0, of the convex hull of 0 and the columns of A. Similarly we obtain Σ L A = {1, 2} for L 3 > 2, which can be read as a convex hull as before, but with a 3 not in the picture. For L 3 = 2, gr L I A = I A is prime and Σ L A should now equal {1, 2, 3}: we would like to view a 3 as "collinear with a 1 , a 2 " in this case. This is the topic of the next section; the following is a teaser: in order to view the three cases from a unifying angle, note that scaling a weight component L i by λ and "scaling the degree a i of ∂ i " by 1/λ have the same effect on the initial terms (and also on the face complex of Σ L A ). One is thus lead to replace a 3 by a 3 /L 3 ; then the resulting convex hull yields the face complex generated by We denote the L-leading term of P ∈ D A by σ L (P ) and call it the L-symbol.
Convention 3.4. We assume that there is a positive real constant c such that This hypothesis has the effect that is a (commutative) polynomial ring whose spectrum is naturally identified with the total space of the cotangent bundle T * C n of C n . Moreover, each E i is Lhomogeneous of positive degree. The W A -ideal gr L (J) defines the L-characteristic variety ChV L (M ) of the module M ; for a holonomic module M it is purely n-dimensional by a result of G.G. Smith [Smi01].
We record the special case Our plan is to connect this construction to analytic information as follows. Suppose X ′ ⊆ X = C n,an is an analytic subspace with a smooth point x ∈ X ′ . Then in suitable local coordinates at x one can write X ′ as the zero set of the first n−dim X ′ coordinates on X. In the stalk at x consider the grading of the D-module M by the filtrations induced by the weights L p/q := pF + qV where as always F is the order filtration and V is the V -filtration along X ′ (compare Subsection 3.1): (There is an obvious identification of graded objects for L p/q and L p ′ /q ′ when p/q = p ′ /q ′ ).
Definition 3.5. With notation as just introduced, p/q ∈ Q is a slope of M along X ′ if ChV L (M ) = supp(gr L (M )) jumps at p/q. This means that ChV L ε (M ) is for small ε ∈ R + constant on (−ε + p q , p q ) and ( p q , p q + ε) but not on (−ε + p q , p q + ε). ♦ This definition is taken from [Lau87]. By [LM99], Laurent's algebraic slopes constructed from filtrations agree with Mebkhout's transcendental slopes given as jumps of the Gevrey filtration on the irregularity sheaf and hence provide a measure of growth for the solutions of M . The central question in this section is to study the behavior of ChV L (M A (β)) under changes of L and β.
We illustrate the link of slopes of M A (β) with Fuchs' criterion in an example.
Example 3.6. It is clear from the series expansion (2) that the Kummer confluent series 1 F 1 (a; b; z) is analytic at every finite z for all a, b. On the other hand, it follows from the integral definition of the error function that at z = ∞ there is an essential singularity (and algebraic changes of coordinates do not eradicate essential singularities). If we denote −1/z by u, then the differential operator θ z (θ z + 1/2) − z(θ z − 1/2) turns into uθ u (θ u − 1/2) − (θ u + 1/2) for the resulting inverse Kummer confluent series.
The For the translation to the A-hypergeometric setting we can use in both cases A = 1 0 1 0 1 1 , with v being (1, 1, −1) or (−1, −1, 1). The toric ideal is then We know from Example 3.1 that for the family L t = (1, 1, t) there is a jump at t = 2 in the L t -graded ideal of I A since at that moment v becomes L-homogeneous. It turns out that the L t -characteristic variety of H A (β) for any β also changes at t = 2, so that M A (β) has a slope of 2 along the hyperplane x 3 = 0.
The correspondence between these numbers is encapsulated by the equation 1 sF = 1/sL 1/sL−1 , where s F is the slope of the Fuchs polygon (and indicates exponential growth behavior with exponent s F ), and s L is the slope at which Laurent's filtrations jump. ♦ We now discuss "regular triangulations to non-monomial graded toric ideals" coming from non-generic weight vectors in greater generality, the details being taken from [SW08]. For the transition, suppose J is generated by elements inside R A ⊆ D A . Then one can restrict the weight to L ∂ on R A and compute gr L ∂ (J ∩R A ) in the commutative situation of Subsection 3.2. Note that then gr L (J) = gr L (D A ) · gr L ∂ (J ∩ R A ). Specifically, we write . . . , L n ) ∈ Q n be any weight vector on R A . As L may have zero components, possible division (as suggested in Example 3.3) by L i = 0 forces us into work in a projective space: In P d Q , any two distinct points a, b ∈ P d Q are joined by two line segments. If the hyperplane H in P d Q contains neither a nor b, one may define the convex hull of a, b as the as the line segment not intersecting H. Similarly one can define the convex hull conv H (S) of a subset S ⊆ P d Q disjoint from H as the convex hull of S in the affine space P d Q H. Example 3.8. Figure 7 shows the (A, L)-umbrella for the matrix A = 1 0 1 2 0 1 1 3 for various filtrations in the family L t = (1, 1, 1, t). While moving the parameter, Φ L A jumps exactly at t = 2 and t = 3. For the intervals t < 2, t = 2, 2 < t < 3, t = 3, t > 3, the corresponding complexes Φ Q are in bijection with those of the cone spanned by it from the origin in A d+1 Q that have outer normal vector "pointing down", and this is the same cone as the one spanned by the appropriate collection inside {(a j , L(a j )} n 1 . ♦ Just like Σ L A in the monomial case, Φ L A corresponds to minimal prime ideals of gr L (I A ). More precisely the following holds.
Theorem 3.10 ([SW08, Thm. 2.14]). The set of A-graded prime ideals containing In particular, the (A, L)-umbrella encodes the geometry of S L A . 3.4. L-characteristic varieties. Equipped with the knowledge from the previous section, we can return to the question of describing . For a weight L ∈ Q n × Q n , the L-symbols σ L (E i ) span the tangent spaces of every torus orbit and hence impose the conormal condition to O τ A for all τ ∈ Φ L A (compare [GZK89,SW08]). The inclusion (16) gr L (H A (β)) ⊇ σ L (E) + gr L (D A · I L A ) appears already in [GZK89,Ado94] and shows that ChV L (M A (β)) must be contained in the union of the closures of all these conormals.
One might hope that (16) is always an equality; this would simplify the problem of describing ChV L (M A (β)). The right hand side is the fake initial ideal and equality holds if I L A is Cohen-Macaulay, [SST00, Thm. 4.3.8]. Unfortunately, this inclusion can be strict in general as the following example shows.
Notwithstanding this example, the following is true.
Theorem 3.12. The L-characteristic variety of the A-hypergeometric system is where for τ ∈ Φ L A , we denote by Υ τ A ⊆ T * C n the conormal to the orbit O τ A ⊆ C n , and where we use the identification T * C n ∼ = T * C n .
By Theorem 3.12 the two ideals in (16) differ along minimal components only by their multiplicities. Taking into account this information turns the L-characteristic variety ChV L (M A (β)) into the L-characteristic cycle ChC L (M A (β)) of M A (β). Let µ L,τ A,0 (β) be the multiplicity of Υ τ A in ChC L (H A (β)). This number is bounded from below by the intersection multiplicity µ L,τ A of the Euler variety Var(gr L (E 1 , . . . , E d )) ⊆ C n with the component of gr L (I A ) along Υ τ A . Moreover, it agrees with this estimate for a Zariski-open set of parameters β, but may exceed it for special values of β, see [SW08].
Using this notation, with volume functions normalized such that they return unity on the standard simplex, In particular, this formula proves that the slopes of the D-module M A (β) are determined entirely by combinatorics of A L , since this is true for their L-characteristic varieties. (For the empty face τ , if NA is saturated, this simplifies to the formula already in [GZK89] that rank is then equal to the volume of A).
Remark 3.13. If an A-hypergeometric system is homogeneous, it can have no slopes since it is regular holonomic [Hot98]. On the other hand, an inhomogeneous H A (β) has at least one slope along the subspace cut out by the variables corresponding to any of the faces of the umbrella of A that do not touch the boundary of the umbrella, as moving it will eventually change the shape of the umbrella. By Laurent's results, regularity of M A (β) is hence equivalent to homogeneity and independent of β. ♦ Remark 3.14. A natural question is whether one can find a stratification of the parameter space such that rank is constant on each stratum and one can give a family of parametric solutions that deform analytically to rank many solutions on the chosen stratum. This is indeed so, the details are worked out in [BFP14,BZFM16b,BFM18]. For confluent systems, when the Nilsson ring does not contain all solutions, the approach of Gevrey series can be used. Early focus was on the irregularity sheaves of Mebkhout introduced in [Meb90]. In a series of papers, Castro-Jimenez and Fernandez-Fernandez [FF10,FFCJ11b,FFCJ11a,FFCJ12], study theory and construction of solutions. Another point of interest is asymptotics. In [CJG15] it is worked out how this plays out in the d = 1 case (A is a single row matrix): Gevrey series solution along the singular locus of the system appear as asymptotes of holomorphic solutions along suitable paths of integration. A similar result for modified systems is proved in [CJFFKT15].
A related problem is that of determining the monodromy of A-hypergeometric systems. This turns out to be an extraordinarily difficult problem, and only limited information is available at this point. We mention the work of Ando, Esterov and Takeuchi [AET15] that determines the monodromy at infinity for confluent (inhomogeneous) systems, building on [Tak10] for the homogeneous case. Hien's rapid decay cycles ( [Hie09]) make an entry here via [ET15], replacing the classical integral representations of Gel ′ fand et al. ♦

Hodge theory of GKZ-systems
In this section we show that certain GKZ-systems carry a mixed Hodge module structure in the sense of [Sai90] and investigate some consequences of this fact.
Since the definition of mixed Hodge modules (MHM) is rather involved, we give here a simplified version which is enough for our purpose. Assuming the reader to be at least somewhat acquainted with the Riemann-Hilbert correspondence, we start with a brief outline of the cornerstones of the theory of mixed Hodge modules. We then give (certain) A-hypergeometric systems an interpretation as Gauß-Manin systems and derive from that a MHM structure. We then discuss two induced filtrations on these GKZ-systems. The standard example of a (mixed) Hodge module on a smooth variety X is the structure sheaf O X : it carries a canonical mixed Hodge module structure, which satisfies Our starting point is section 2.3, where we have seen that if β ∈ sRes(A) then which lift the corresponding functors f + , f † , f + , f † on the category of regular holonomic D-modules. So, in particular, if O β T is in MHM(X) then so is M A (β) whenever β ∈ sRes (A).
In order to have M A (β) be a mixed Hodge module, it should of course in particular be regular holonomic. By Remark 3.13 and Definition 1.7, this is equivalent to I A being homogeneous. In other words, we must require that the vector (1, 1, . . . , 1) is in the row span of A. This required homogeneity of A coincidentally provides the solution to an issue not mentioned yet: the (inverse) Fourier transform does in general not preserve mixed Hodge modules. In order to construct a mixed Hodge module structure on a GKZ-system via M A (β), we use a Radon transform, which does carry mixed Hodge structures and which only makes sense in the homogeneous context.
In order to simplify the statement of some formulas in the remainder of the article, we make now the following convention on A.
Convention 4.1. From now on, A is in Z (d+1)×(n+1) and we assume that A is homogeneous, full, pointed, and generates a saturated semigroup. ♦ Since a GKZ-system derived from a pair (A, β) is unchanged under an invertible Z-linear transformation of the rows we can moreover assume that the matrix A has the following shape where B ∈ Z d×n is full but is not necessarily pointed or homogeneous. Notice also that if NA is saturated, then so is NB, however, the converse implication is not true in general.

Geometric interpretation of GKZ-systems. The aim of this section is
to express certain GKZ systems as objects which are built from consecutive applications of (possibly proper) direct image and (possibly exceptional) inverse image functors applied to a structure sheaf. From the discussion above it follows then that these GKZ systems carry a mixed Hodge module structure. In order to achieve this we have to introduce various integral transformations and their relations.
Define a pairing y j x j , and a free rank one O C n+1 ×C n+1 -module which acquires a D C n+1 ×C n+1 -module structure via the product rule. We denote by p 1 and p 2 the projections from C n+1 × C n+1 to the first and second factor respectively. The sheafified version of the Fourier transform is given by and one has FL • FL = − id. Although defined at the level of derived categories, FL is an exact functor, and an instructive exercise shows that on the level of global sections it is given by formula (12). Theorem 2.10 now implies that, whenever β ∈ sRes(A), we have Here, the final identification holds due to the homogeneity of I A even though FL 2 is not the identity.
The second type of transformation we will need is the Radon transformation of D-modules introduced by Brylinski [Bry86]; some variations were later discussed by d'Agnolo and Eastwood [DE03]. Let be the complement of the universal hypersurface defined by the vanishing of the pairing −, − . For the sake of readability, we denote P( C n+1 ) form now on simply by P n . Consider the following commutative diagram The Radon transformation is the functor RT :  rh (D P n ). There are the following isomorphisms where j : C n+1 \ {0} ֒→ C n+1 is the canonical inclusion.
In particular, if N is a mixed Hodge module, then the above isomorphisms allow us to equip the right hand sides with induced MHM structures.
To simplify the presentation, we will focus now (and this until Definition 4.5 below) primarily on the case β = 0. For β = 0 a twisted variant of the Radon transformation is needed: see [RS20] for details. We start with the following commutative diagram In particular, h A : T −→ C n+1 is as in (15) earlier (with the caveat that now A is as in Convention 4.1). We then observe that We now consider a part of the long exact sequence of the adjunction triangle (20) applied to (g B ) + O T . In order to identify the individuals terms we introduce a family of Laurent polynomials defined on (C * ) d × C n = T × C n using the columns b 1 , . . . , b n of the matrix B from (17). We define

Theorem 4.3 ([Rei14, Cor. 2.3]).
There is the following commutative diagram with exact rows where all vertical maps are all isomorphisms; just for this statement we abbreviate for typesetting reasons g B by g and denote the Radon transform by just R.
As a consequence, the lower exact sequence underlies a sequence of mixed Hodge modules.

Hodge-filtration on GKZ-systems.
Although the isomorphism (23) equips the GKZ system M A (0) with the structure of a mixed Hodge module, it is far from clear what the Hodge and weight filtrations look like. The first step in this direction was carried out by Stienstra [Sti98], relying heavily on work of Batyrev [Bat93], who computed the Hodge and weight filtration on the smooth part of the GKZ system. Denote ∆ := conv(a 0 , . . . , a n ) the convex hull of the points a 0 , . . . , a n , and note that this is the decone of the A-polyhedron from Definition 3.7. Let τ ⊆ ∆ be a face of ∆, let x ∈ C n , and set have no common solutions in T. Then, for 0 ≤ i ≤ d, define the differential operators which are elements of the Weyl algebra D C[t ± ] on t 0 , . . . , t d localized at t 0 · · · t d . One checks that these operate on the semigroup ring S A ⊆ C[t ±1 0 , . . . , t ±1 d ], P i (S A ) ⊆ S A , so they are differential operators on the affine toric variety X A = Spec(S A ).
Before we can state Stienstra's result mentioned in the introduction to this section, we need some more terminology. Let ∆ is generated by all elements t a with a ∈ NA that are not contained in any codimension k face of R ≥0 A. Define a decreasing sequence of C-vector spaces in S A Stienstra proved the following result recall that ϕ is the family defined in (24). Under this isomorphism, the Hodge filtration is given by If the matrix B ∈ Z d×n is homogeneous, then the weight filtration on H d (T, ϕ −1 (x); C) is given by where the semigroup ring S B , the ideals I (k) ∆ and the differential operators P i are now derived from B.
Notice that equation (26) is shown in [Sti98] only for the case where A is homogeneous, the general case is treated in [RS20].
The Since we will formulate the result for certain parameter vectors β different from 0, we first need to introduce the following definition.
Definition 4.5. The set of admissible parameters β ∈ R d+1 ⊆ C d+1 is defined by where ε A := a 0 + . . .+ a n , e τ := n τ , ε A ∈ Z >0 and n τ is the unit, inward pointing, normal vector of τ . ♦ Example 4.6. For the matrix A = 1 1 1 1 0 −1 1 2 , the following picture shows the sets sRes(A) (see Definition 2.6 above) and A A . ♦ We can now state a result, taken from [RS20, Theorem 5.35] which describes the Hodge filtration on the GKZ-systems in a rather precise way.
Theorem 4.7. Let A ∈ Z (d+1)×(n+1) be as in Convention 4.1, β ∈ A A and β 0 ∈ (−1, 0]. Then the Hodge filtration on M A (β) is given by the shifted order filtration, i.e. we have the following equality of filtered D C n+1 -modules It has been shown in [RS20,Theorem 5.43] that the first part of the above Theorem 4.4, i.e. Formula (26) is a rather direct consequence of the comparison between the Hodge and the order filtration on M A (0). Remark 4.8. As already noted in Section 2 above, a variant of Borisov-Horja's better behaved GKZ-systems has been considered in [Moc15b]. If we suppose that A is normal (as we do throughout this section), then the definition in [Moc15b] coincides with the one for ordinary GKZ-systems as given in 1.6 above. However, the matrix A is not supposed to be homogeneous in [Moc15b]. The module M A (β) will have irregular singularities then, as discussed in Section 3 above. One may ask what kind of Hodge theoretic information can be derived from M A (β) in this case. This is similar to the statements on the ordinary versus irregular Hodge filtration on univariate hypergeometric systems that we will discuss below.
In [Moc15b, Prop. 1.4], Mochizuki proves the the following statement which can be considered as an irregular variant of Theorem 4.7 above. Let B ∈ Z d×n be such that ZB = Z d . Suppose for the simplicity of the exposition that NB = R ≥0 B ∩ Z d . Consider the non-commutative "Rees ring" and the corresponding sheaf R C×C n . Let H z A (0) be the left R C×C n -ideal generated by Then the left R C×C n -module R C×C n /H z A (0) underlies a mixed twistor module on C n , a notion that in many respects is the correct replacement of a mixed Hodge module in the irregular setup. In particular, any mixed Hodge module can be considered as a special mixed twistor module, and therefore the case β = 0 of Theorem 4.7 can be deduced from Mochizuki's result. Using a filtered variant of the Fourier-Laplace transformation (compare the discussion in Section 5 below), one can also obtain the latter from Theorem 4.7, as has been demonstrated in [CnDRS19, Corollary 4.8]. ♦ As another application of Theorem 4.7, we will describe some results about the Hodge structure of univariate hypergeometric equations (see the discussion in Subsection 1.2 above). Consider again the operator (compare with equation (7), where m ′ = q + 1, m = p and where λ 1 = 0, λ i = 1 − β i+1 , µ j = −α j ) for some real numbers λ i , µ j . The corresponding cyclic module is irreducible if and only if for all i, j we have λ i − µ j / ∈ Z. The modules H (λ; µ) are the most basic examples of rigid D-modules (see [Kat90,Ari10]). A first consequence of this property is that if H (λ; µ) is irreducible, then it is isomorphic to some H (λ ′ ; µ ′ ) whenever µ − µ ′ and λ − λ ′ are integer vectors. We can thus assume that 0 ≤ λ 1 ≤ . . . , λ m ′ < 1, 0 ≤ µ 1 ≤ . . . ≤ µ m < 1 and that λ i = µ j for all i, j. It is obvious that H (λ; µ) is regular exactly when m ′ = m and in that case it has the three singular points {0, 1, ∞}. On the other hand, if m ′ = m then Sing(H (λ; µ) = {0, ∞}.
In the regular case, that is, if m ′ = m, the rigidity property can be stated at the level of the the local system L on P 1 \{0, 1, ∞} of solutions of P : it simply says that the local monodromies around the singular points determine the (global) monodromy representation defined by L . From there it follows by [Sim90,Cor. 8 The Picard-Fuchs equation of the family of elliptic curves in Example 1.3 corresponds, as we computed there, to the hypergeometric differential equation given by the module H (0, 0; 1/2, 1/2). Applying Fedorov's formula yields dim(gr F 0 L ) = dim(gr F 1 L ) = 1, confirming our computation in Example 1.3. Notice also that in this case the local system L underlies a real (and even rational) variation of Hodge structures, which is consistent with [Fed18, Theorem 2].
If m ′ = m (and, up to a change of the coordinate z → 1/z we can assume that m ′ > m), then H (λ; µ) is irregular and can no longer support a variation of Hodge structures. In [Sab18], a category of irregular Hodge modules is developed, which can roughly be seen as lying between the category of mixed Hodge modules and the category of mixed twistor modules. A possibly irregular D X -module M on a complex manifold X underlying an irregular Hodge module comes equipped with an irregular Hodge filtration, an increasing filtration F irr α M by coherent O Xmodules indexed by the real numbers (contrarily to the regular case); we write F irr <α M := β<α F irr β M . However, the indexing set is determined by a finite set I ⊆ [0, 1) having the property that In [SY19], the following formula for the irregular Hodge numbers has been found (see also [CDS19] and [CnDRS19], where the Hodge filtration itself is determined in some cases, using Theorem 4.7 from above): For m ′ = m, this gives back the formula (31) up to the fact that the local system L is in the regular case in [Fed18] the one of the solutions of H (λ; µ), whereas formula (32) gives (for m ′ = m) Hodge numbers of a filtration defined on the dual local system of flat sections.
x x r r r r r r r r r r r where p 1 is the projections to the first factor, i 0 is the canonical inclusion and the map ω is given by The Fourier-Sato transformation (or monodromic Fourier transformation) is defined by where φ z is the vanishing cycle functor along z = 0.
It was shown in [RW,Proposition 4.12] that the Fourier-Sato transformation respects the weight filtration of monodromic D-modules which are localized along {0} ∈ C n+1 (up to a shift). Hence, a weight filtration on the GKZ-system is induced by the following isomorphisms: Since the Fourier-Sato transform is an equivalence of categories it is therefore enough to compute the weight filtration on M A (0) = (h A ) + O T which will be done below.
Recall that the graded parts Gr W k M of a mixed Hodge module are pure Hodge modules and as such are semi-simple, i.e. they are direct sums of intersection complexes. Because the number of simple objects (counted with multiplicity) is independent on the chosen (weight) filtration this also gives us the simple objects occurring in the weight filtration induced by the Radon transform (but possibly in another order). However, we conjecture that the Fourier-Sato transformation and the Radon transformation are isomorphic on the level of mixed Hodge modules, i.e.
Conjecture 4.9. For N ∈ MHM(P n ): We will now proceed to state the result on the weight filtration of Let τ ⊆ γ ⊆ σ be faces of a cone σ ⊂ R d+1 . The quotient face of γ by τ is defined as: The cone γ ℧ is the dual of γ in its own span, hence independent of σ. For cones τ ⊆ γ denote by X γ/τ the spectrum of the semigroup ring induced by the cone γ/τ in its natural lattice. Set Y γ/τ := X (γ/τ ) ℧ .
In the following, we denote the cone R ≥0 A by σ. The Fourier transformed GKZ system M A (0) is isomorphic to (h A ) + O T and has support on the affine toric variety X A = X σ . For a face τ of σ write d τ for its dimension. We have seen in Subsection 2.1 that the d τ -dimensional T-orbits O τ A in X σ are in one-to-one correspondence with the faces τ of σ. The closure of an orbit O τ A is X τ . It turns out that the varieties X τ are exactly those which occur as support varieties of the summands in the semisimple decompositions of the graded parts gr W M A (0).
Let L (τ,d+e) be the constant local system of rank dim IH d+1−dτ −e (Y σ/τ ) on O τ A . In order to simplify the notation, we use the symbol IC Y (L ) for the intersection cohomology D-module on some smooth variety X with support on the closed subset Y ⊆ X, and where L is a local system on a Zariski open subset of Y .

Application to toric mirror symmetry
The aim of this final section is to discuss some results concerning the so-called mirror symmetry phenomenon, which links enumerative geometry of projective algebraic, and more generally symplectic varieties (called A-model) to complex geometry, in particular, Hodge theory of their so-called B-models. The B-model is usually given by a family of algebraic varieties which may have singularities and which need not be projective (which forces one to consider compactifications, see below). Often these families on the B-side are referred to as Landau-Ginzburg models.
The first example of mirror symmetry was given by Candelas, de la Ossa, Green and Parkes [CdlOGP91] who predicted a virtual number of rational curves on a quintic threefold (later referred to as the genus 0 Gromov-Witten invariants) by period computations for the mirror partner, i.e. the B-model. These predictions were verified and also generalized to numerically effective smooth complete intersections in toric varieties by Givental [Giv96], [Giv98]. His celebrated mirror theorem shows that the J-function, a generating function for the genus 0 GW-invariants of such varieties, is computable in terms of a cohomology-valued hypergeometric function. Givental also conjectured that the components of this function are given as oscillating integrals. This was much later proved by Iritani in [Iri09] (even treating the case where the toric variety in question is an orbifold), some details of the construction described below are parallel to his paper. However, an algebraic construction of the correct Hodge theoretic B-model was still missing. Our purpose in this section is to give an overview of techniques and results (mainly referring to [RS15,RS17,RS20] as well as to [Moc15b]), where the machinery of GKZ-systems as discussed in the previous sections is used to obtain a purely algebraic Hodge theoretic (and D-module based) mirror correspondence for certain smooth toric varieties resp. subvarieties of them. 5.1. Gromov-Witten invariants and Dubrovin connection. Let X be a toric smooth projective variety. For the purpose of this exposition, we assume further that X is Fano, so the anticanonical class [−K X ] is ample. A good part of the results discussed below also applies if one considers weak Fano manifolds, meaning that [−K X ] is a numerically effective (nef) class. There are however a few technical modifications needed in the nef case, which is why we refrain from discussing it here. Developing the mirror symmetry picture described below in the absence of any positivity assumption on X remains a subject of active current research (see, e.g., [Iri08], [GKR17], [Iri17]).
Let β ∈ H 2 (X, Z) and choose γ 1 , γ 2 , γ 3 ∈ H * (X, Q). The genus zero, three point Gromov-Witten invariants intuitively count the number of stable maps f from rational curves C with-in this case-three marked points, satisfying f * ([C]) = β and f (C) ∩ PD(γ i ) = ∅ for i = 1, 2, 3. (Here and elsewhere, PD(−) denotes the Poincaré dual). Technically, they are obtained as follows: pull back the (three) arguments of I 0,3,β to the moduli space of such maps (along the three induced evaluation maps to X), take their cup product and evaluate against this product by integration over a certain virtual fundamental class on the moduli space. Constructing this latter class is a major issue in Gromov-Witten theory (see, e.g. [FP97] and [BF97]).
We choose a homogeneous basis T 0 , T 1 , . . . , T r , T r+1 , . . . , T s of H * (X; Z) such that T 0 ∈ H 0 (X; Z), the classes T 1 , . . . , T r ∈ H 2 (X; Z) lie in the nef cone of X and T r+1 , . . . , T s ∈ H >2 (X; Z). Let g ij := (T i , T j ) be the Poincaré pairing between the elements T i and T j and define With δ ∈ H 2 (X; C), the three point Gromov-Witten invariants can be used as structure constants for a family of multiplications on H * (X; C). This product structure is the small quantum product of X and parameterized by the cosets of δ in the complexified Kähler moduli space A priori it is far from clear that the sum in (33) is convergent. However, the Gromov-Witten invariants satisfy (among others) the following properties: Effectivity : I 0,3,β = 0 if β does not lie in the Mori cone Degree : where we recall that the Mori cone is the cone in H 2 (X; R) of effective classes of curves. It is dual to the cone of nef divisors in H 2 (X; R). The effectivity axiom together with our assumption that X is Fano (i.e. that the class c 1 (X) is ample) show that I 0,3,β is zero unless c 1 (X)(β) ≥ 0. The degree axiom now tells us that for fixed T i , T j , T k there are only finitely many β in the Mori cone such that I 0,3,β (T i , T j , T k ) is non-zero. Hence the product defined in (33) is finite and therefore defined on the whole space K. It can be seen from other axioms that the small quantum product is commutative, associative and that T 0 acts as identity. Let η 1 , . . . , η r ∈ H 2 (X, Z) such that T i (η j ) is the Kronecker δ i,j for 1 ≤ i, j ≤ r. If we write δ = t 1 T 1 + . . . + t r T r ∈ H 2 (X; C), β = β 1 η 1 + . . . + β r η r ∈ H 2 (X; C), and set q i := exp(t i ) for i = 1, . . . , r, we get exp(δ(β)) = q β1 1 . . . q βr r . Then, under the exponential map from H 2 (X; C) to K, q = {q i } i=1,...,r become coordinates on K corresponding to t = {t i } i=1,...,r on H 2 (X; C) and induce an explicit isomorphism K ≃ (C * ) r . Since T 1 , . . . , T r lie in the nef cone, the cone generated by the dual basis (η j ) j=1,...,r contains the Mori cone and therefore all monomials q β1 1 . . . q βr r have non-negative exponents. Hence the quantum product extends to the partial compactification (34) K := C r ←֓ (C * ) r = K.
The point mapping property of the Gromov-Witten invariants shows that the small quantum product degenerates to the ordinary cup product at q = 0.
Example 5.1. Consider the first Hirzebruch surface F 1 which is induced by the following fan (left); on the right is shown the space H 2 (F 1 ; R) using the coordinate system given by the classes of D 1 and D 2 . (See the start of Subsection 5.2 for information on how to view H 2 (X; Z)).
The small quantum cohomology ring of F 1 is therefore given by C[q 1 , q 2 , T 1 , T 2 ]/ T 2 1 + q 1 T 1 − q 2 T 2 , T 2 2 − T 1 T 2 − q 2 , T 1 T 2 2 − q 2 T 1 − q 1 q 2 . Restricting this ring to q 1 = q 2 = 0 gives C[T 1 , T 2 ]/(T 2 1 , T 2 2 − T 1 T 2 , T 1 T 2 2 ) which is isomorphic to the cohomology ring (cf. [Ful93, Section 5.2]), We are going to give a reformulation of the quantum cohomology algebra in terms of certain differential systems. The intrinsic reason of the appearance of differential equations in this context is best understood when studying the big quantum product instead of the small one as we have done above. It basically means to have a product on H * (X; C) which is parameterized by any class δ ∈ H * (X; C) instead of a class in H 2 (X; C) (more precisely, instead of a representative of a coset in K). One can show that the structure constants of the big quantum product can be obtained as third derivatives of a generating function, referred to as the Gromov-Witten potential. This fact reveals an intrinsic integrability property of the (big) quantum product. Moreover, the associativity then boils down to a famous third order non-linear partial differential equation satisfied by the GW-potential, abbreviated as WDVVequation (after Witten, Dijkgraaf, Verlinde, Verlinde, see, e.g. [Man99]). It turns out that using the next definition, this equation can be rewritten as a flatness property of a system of linear differential equations, that is, a vector bundle with a connection.
Definition 5.2. The small Dubrovin connection (H A , ∇ A ) of X is a flat meromorphic connection ∇ A on a trivial, holomorphic vector bundle H A over P 1 × K with fiber H * (X; C). The connection is given by where we denote by z the coordinate centered at 0 ∈ C ⊆ P 1 . ♦ Notice however that this convention from quantum cohomology literature leads to some slight clash of notation. Namely, the variable z from above (a coordinate on P 1 ) is different from the variable z used for univariate hypergeometric equations in Section 1 as well as in Formula (30). In order to be consistent with the literature, we stick to these conventions and hope that it does not lead to confusion.
It is an easy but instructive exercise to check that the flatness of the connection ∇ A implies the assocativity and commutativity of the small quantum product.
Example 5.3. The small Dubrovin connection of the first Hirzebruch surface is given by of algebraic tori, where b is the monomial map encoded by the transpose of B, K is as in Subsection 5.1, and T as in (22). Recall that the standard basis e 1 , . . . , e d of M gives coordinates t = (t 1 , . . . , t d ) on T.
The canonical basis of torus-invariant divisors D 1 , . . . , D n for Div T (X) corresponding to the one-dimensional cones induces an isomorphism Div T (X) ⊗ Z C * ≃ (C * ) n . Let W : Div T (X) ⊗ Z C * = (C * ) n −→ C be the function given by summing the coordinates.
Definition 5.4. The Landau-Ginzburg model associated to the smooth, toric, Fano variety X is the map ♦ If we view K as an abstract algebraic torus, defining the morphism (W, c) requires only the matrix B (that is, the generators of Σ X (1)), but not the full data of the fan Σ X . We shall later wish to (partially) compactify K, as we have done before (see Formula (34)). For this, we need to equip K with the coordinate system {q i } i=1,...,r , corresponding to the basis {T i } i=1,...,r on H 2 (X; C). The compactification is designed to contain the point q 1 = . . . = q r = 0, since there the quantum product collapses to the cup product. This will be the case if the basis {T i } i=1,...,r of H 2 (X; R) consists of nef classes (this choice has already been made above at the beginning of Subsection 5.1). Hence, fixing such a good coordinate system {q i } i=1,...,r on K depends on the geometry of the toric variety X Σ and not just on the ray generators given by the matrix B (see [RS15,Section 3.1] for a more detailed discussion).
Since (37) splits, we can find a section of the map Div T (X) −→ H 2 (X, Z) which then induces a section Again, s, seen as a monomial map from (C * ) r to (C * ) n , will depend on the fan structure of Σ X via the choice of coordinates on K. From now on, we will always fix such coordinates and consider K as the concrete r-dimensional torus (C * ) r . Example 5.5. We continue Example 5.1. The exact sequence (37) is given by The corresponding family of Laurent polynomials is where we have chosen the section s : K −→ Div T (X) ⊗ Z C * as the one induced from the map It was conjectured by Givental (see, e.g. [Giv98]) that oscillating integrals over Lefschetz thimbles with respect to the Landau-Ginzburg model give flat sections of the Dubrovin connection. An algebraic replacement of these oscillating integrals, localized and partially Fourier transformed Gauß-Manin systems of the Landau-Ginzburg model.
We briefly explain this version of the ordinary Fourier transformation functor (see Formula (19) above). In the following, O Ct×Cτ ×Y · exp(−tτ ) denotes a free rank 1 module with twisted differential given by the product rule.
Definition 5.6. Given a smooth variety Y and a holonomic D C×Y -module N , the localized, partial Fourier transform of N is the sheaf are the canonical open embeddings with the understanding that z = 1/τ . ♦ The name "localized" comes from the fact that by using the direct image (j z ) + , the action of z is invertible on the resulting module (and so is the action of τ ).
The localized, partial Fourier transformed Gauß-Manin system of the Landau-Ginzburg model ψ is then defined as . It is an exercise (using the definition of the direct image functor, see, e.g. [HTT08, Sections 1.3, 1.5]) to show that the module of global sections G ψ of G ψ has the following presentation in terms of relative differential forms where d is the differential on the complex Ω •+d T×K/K . Following an idea from singularity theory (see [Bri70,Sai89,Sab06]), one defines the Fourier transformed Brieskorn lattice by We will see below, using GKZ-systems, that G ψ 0 is O C×K -free. In order to connect G ψ to a GKZ-system we observe that the family of Laurent polynomials ψ is a pullback of a larger family x j t bj , (x 1 , . . . , x n )) by the map where s : K ֒→ Div T (X) ⊗ Z C * ∼ = (C * ) n is as in (38) and the middle map is the identification induced from the standard basis on M . In Theorem 4.3 we have connected the Gauß-Manin system of ϕ to a GKZ system via the 4-term sequence is the homogenization of the matrix B constructed from the ray generators of the fan Σ X . Since the outer two terms are free O C n+1 -modules, they are in the kernel of the localized partial Fourier transform. Indeed, on the level of global sections, FL loc Y is the composition the localization at ∂ t with the ordinary Fourier transformation FL Y , and C[t] = D t /D t ·∂ t naturally localizes to zero. Thus, the localized Fourier transform being the composition of two exact functors, the previous display implies The module of global sections of FL loc C n (M A (0)) is the cyclic left module D C×C n [z ± ]/I over the ring where I is generated by the operators E 0 , ( E i ) i=1,...,d , ( u ) u∈ker(B) from Equation (29). We like to compare this computation to a presentation for the Fourier transformed Brieskorn lattice G ϕ 0 ⊆ G ϕ for the map ϕ instead of ψ. For this, we use again the Rees ring R C×C n = C[z, x 1 , . . . , x n ] z 2 ∂ z , z∂ x1 , . . . , z∂ xn from Equation One can prove by base change that the Fourier transformed Brieskorn lattice G ϕ 0 is the inverse image of G ψ 0 under the map ι in (42). We therefore arrive at the following result where, for u ∈ ker(B), we read it as an element of H 2 (X; C) via the dual of the sequence (37): Parallel to R C×C n from (28), we define R C×K := C[z, q ± 1 , . . . , q ± r ] z 2 ∂ z , z∂ q1 , . . . , z∂ qr and denote by R C×K the associated sheaf on C × K.
Proposition 5.8. The localized Fourier transformed Brieskorn lattice G ψ 0 is O C×Kfree. As a sheaf over R C×K , it is isomorphic to the cyclic module R C×K /J where the left ideal J is generated by (here, u runs through ker(B) and {q a } a=1...,r are coordinates on K as always) R log C×K := C[z, q 1 , . . . , q r ] z 2 ∂ z , zq 1 ∂ q1 , . . . , zq r ∂ qr and denote by R log C×K the associated sheaf on C × K. Then the following statements on some cyclic R log C×K -modules are proved in [RS15] using methods from toric geometry, including the notions of primitive collections and relations (see, e.g., [CvR09,CLS11]).
Proposition 5.9. Let J log ⊆ R log be the left ideal generated by E and u from Proposition 5.8. Then In order to construct an object which matches the small Dubrovin connection coming from the Gromov-Witten invariants of X we have to go one step further.
Recall that the small Dubrovin connection (35) is a family of vector bundles on P 1 , parameterized by K, equipped with a certain connection operator. As of yet, starting from the Landau-Ginzburg model ψ from (39) of X, we have constructed a vector bundle R log C×K /J log on C × K with a differential structure, and it is easily verified that the behavior along the poles ({0} × K) ∪ (C × (K\K)) of the connection operators on both bundles are of the same type. If we want to compare R log C×K /J log to the small Dubrovin connection, it thus remains to extend this bundle (together with its connection operator) over the divisor {∞} × K to all of P 1 × K. This is of course always possible if no other condition is imposed. However, if we want to reconstruct the Dubrovin connection, this extension needs to satisfy two strong conditions simultaneously: the resulting object must be a family of trivial P 1bundles and the connection must have a logarithmic pole at infinity. Fulfilling both requirements is not always possible, and goes under the name (Riemann-Hilbert-)Birkhoff problem; for a modern account see [Sab98,Chapter IV]. However, under the current circumstances, a solution to the Birkhoff problem can be found locally near the boundary K\K, as the following result shows.
Theorem 5.10. ([RS15, Proposition 3.10]) There exists a Zariski open neighborhood U of 0 ∈ K and sections Q 0 , . . . , Q s of (R log C×K /J log ) |C×U which extend (R log C×K /J log ) |C×U as a (trivial) holomorphic vector bundle over P 1 × U , called H B , such that the associated connection ∇ B has a logarithmic pole along the normal crossing divisor ({z = ∞} × U ) ∪ (P 1 z × (K \ K)). With all these preparations, we can state the following result, which can be considered as the Hodge theoretic mirror statement for smooth toric Fano varieties. Example 5.12. When X is the Hirzebruch F 1 surface, the Fourier transformed Brieskorn lattice of the Landau-Ginzburg model is given by ] z 2 ∂ z , z∂ q1 , z∂ q2 /J where the left ideal J is generated by the operators 1, 1, 0), u 3 = (0, 1, 0, 1) generate the integer kernel of B.
The logarithmic extension is equal to C[z, q 1 , q 2 ] z 2 ∂ z , zq 1 ∂ q1 , zq 2 ∂ q2 /J log where J log is generated by the same operators as J.
The basis which solves the (Riemann-Hilbert)-Birkhoff problem is Q 0 = 1, Q 1 = zq 1 ∂q 1 , Q 2 = zq 2 ∂ q2 , Q 3 = (zq 1 ∂ q1 )(zq 2 ∂ q2 ). These sections are identified with the sections T 0 , T 1 , T 2 , T 3 of H A under the mirror isomorphism from Theorem 5.11. ♦ 5.3. Reduced quantum D-modules and intersection cohomology. In this section, we are going to discuss a mirror statement that concerns weak Fano smooth complete intersections inside smooth projective toric, possibly non-Fano, varieties. From the point of view of physics, this is an even more important class of examples than the one considered previously since it includes Calabi-Yau manifolds that are subvarieties of toric manifolds, although they are not toric themselves. The most prominent example, namely, the quintic in P 4 (where the first enumerative predictions using the mirror symmetry principle were made, see [CdlOGP91]) is of this type. We will discuss a non-affine version of the Landau-Ginzburg models introduced above. The mirror statement that we aim for will relate (part of) the quantum cohomology of the complete intersection subvariety to the lowest weight filtration step of a GKZ-system. It follows from the results in Section 4.3 that the lowest weight filtration step is a single intersection cohomology D-module which arises as the image under a natural morphism from the holonomic dual of the GKZ system to the GKZ system itself. In the cases we discuss here this holonomic dual is isomorphic to a GKZ system with the same matrix A but different parameter vector β. Hence the intersection cohomology D-module can be described as the image of a morphism between two GKZ-systems by a contiguity morphism. Our main reference in this section is [RS17]. We start with setting the notation.
Notation 5.13. As before, X will be a smooth projective toric variety of Picard rank r attached to the fan Σ X of dimension d, whose primitive rays form the columns of the matrix B. In contrast to the previous case we do in this subsection not make any positivity assumption on X here. Let O(L 1 ), . . . , O(L c ) be globally generated line bundles; since X is toric, this amounts to asking that each L i be nef-their classes should lie in the nef cone in H 2 (X, R). We shall assume also that If D 1 , . . . , D n are the torus invariant divisors on X we can write for suitable non-negative integers d ij . Set and consider a generic global section γ ∈ Γ(X, E ). Our assumptions imply that is a smooth complete intersection subvariety for which −K Y is nef; we call this property weak Fano. ♦ In this paragraph we briefly review a variant of the above quantum product that is designed to encode enumerative information about stable maps to Y . The first point is that one can generalize the definition of Gromov-Witten invariants (5.1) to the twisted (three-point) GW-invariants; these are also maps from H * (X, Q) ⊗3 → Q, but Chern classes of certain tautological bundles (on the moduli space of stable maps) derived from E come into play. We denote by I 0,3,β (γ 1 , γ 2 , γ 3 ) ∈ Q the value of such a three point twisted GW-invariant for given cohomology classes γ 1 , γ 2 , γ 3 ∈ H * (X, Q) (see, e.g. [RS17, Section 4.1] for a more detailed discussion, including an explanation for the process γ 3 γ 3 ). Then one defines in complete analogy to Formula (33) the twisted (small) quantum product by γ 1 tw * γ 2 := s a=0 β∈H2(X,Z) where, as before, q are coordinates on K and q β := exp(δ(β)) for β ∈ H 2 (X; C).
We now follow the definition of the small Dubrovin connection, Equation (35), and define the twisted quantum D-module, denoted by QDM(X, E ), as the vector bundle on P 1 × K with fiber H * (X; C) together with the connection given by Notice that, unlike in the Fano case discussed in Subsection 5.2, the convergence of the twisted quantum product is not automatic. We will therefore later restrict to some analytic neighborhood U ⊂ K of the point q 1 = . . . = q r = 0 in K, on which tw * is convergent.
As we are interested in enumerative information about maps to Y := γ −1 (0), the cohomology space H * (X; C) is not a well suited object for a quantum cohomology theory of Y . We therefore consider the Gysin morphism and define the reduced cohomology of (X, E ) to be H * (X) := H * (X)/ ker(m E ).
One checks that the twisted quantum D-module QDM(X, E ) has a quotient bundle QDM(X, E ) with fiber H * (X), and that the connection ∇ tw on QDM(X, E ) descends to QDM(X, E ). We call this vector bundle on P 1 × K with connection (QDM(X, E ), ∇ tw ) the reduced quantum D-module (see [RS17, Definition 4.3] for more details).
We proceed by describing the relevant Landau-Ginzburg models attached to the given data (X, E ). Denote by E ∨ the dual bundle of E , and by its total space. Then V is a (non-compact) toric variety, whose fan Σ V ⊆ (N ⊕ Z c ) ⊗ Z R is given as follows: The set of rays of Σ V are the columns of the matrix where B is the d × n-matrix constructed from the primitive rays in Σ X and where d ji are as in (44). Then the fan Σ V consists of all cones and j 1 , . . . , j ℓ ∈ {n + 1, . . . , n + c}. Notice that we have H 2 (V; Z) ∼ = H 2 (X, Z) ∼ = Z r and that Div T (V) ∼ = Z n+c . Similarly to the discussion in Section 5.2 we then consider a family of Laurent polynomials associated to these toric data.
Definition 5.14. ([RS17, Definition 6.3.]) Let (X, E ) be as in Notation 5.13 and consider the complexified Kähler moduli space K ∼ = H 2 (X; Z) ⊗ Z C * ∼ = H 2 (V; Z) ⊗ Z C * of both X and V. Write T V := (C * ) d+c for the (d + c)-dimensional torus. Then the affine Landau-Ginzburg model of (X, E ) is the morphism where K • ⊆ K is a Zariski open subset on which the Laurent polynomials ψ(−, q) satisfy a nondegeneracy condition (see [RS17, Section 3.2]) and where (s ′ 1 , . . . , s ′ n+c ) ∈ Z r×(n+c) is a section of the projection Div T (V) ։ H 2 (X, Z). ♦ One can establish a mirror symmetry theorem for the twisted quantum D-module which involves the affine Landau-Ginzburg model, very much in the same spirit (without looking at logarithmic extensions over the boundary K\K though, and also neglecting the extension to families of bundles over P 1 ) as Theorem 5.11 above (see [RS17, Theorem 6.13, 6.16] and also [Moc15b]). However, in order to reconstruct the reduced quantum D-module QDM(X, E ), we are forced to look at a compactification of the morphism ψ. In order to define it, consider the map g B ′ : T V = (C * ) d+c ֒→ P n+c (see Formula (22) above). Then define (49) Z • := Γ F to be the closure in P n+c × C × K • of the graph Γ F ⊆ T V × C × K of the function F : T V × K • → C defined in (47). Notice that Z • is a partial compactification of T V × K • , that is, quasi-projective but in general not smooth.
Definition 5.15. Let (X, E ) be as above. Then we call the restriction Ψ : Z • −→ C × K • of the projection pr : P n+c × C × K • → C × K • the non-affine Landau-Ginzburg model of (X, E ). ♦ Clearly, Ψ is a projective morphism, and hence should be considered as a partial compactification of the affine Landau-Ginzburg model ψ.
In a rather similar way to the case of Landau-Ginzburg models of projective toric varieties, we obtain the following description of the relevant Gauß-Manin cohomologies by GKZ-type systems. As a matter of notation, consider the the matrix A ′ ∈ Z 1+d+c,1+n+c obtained by homogenizing the matrix B ′ defined in equation (46), that is due to the special shape of the matrix A ′ . Notice that here we use the coordinates (x 0 , x 1 , . . . , x n+c ) on C × C n+c and ∂ 0 , ∂ 1 , . . . , ∂ n+c for the corresponding partials.
We can now formulate the following statement about the non-affine Landau-Ginzburg.
Theorem 5.16 ([RS17, Lemma 6.4 and Proposition 6.7]). There is an isomorphism of D C×K • -modules where we denote (with a slight abuse of notation) by ι : C × K • ֒→ C × C n+c the embedding already used above (see Equation (42)). Moreover, there is an isomorphism of D C×K • -modules 0)) . Notice that by definition, the intersection cohomology module IC(C TV×K • ) to the constant sheaf on T V × K • becomes a D P n+c ×C×K • -module via Kashiwara equivalence (using the locally closed embedding T V × K • ∼ = Γ F ֒→ Γ F ֒→ P n+c × C × K • ); this is the reason for using the direct image by pr from Definition 5.15. Since it has support on the subvariety Z • , the corresponding perverse sheaf under the Riemann-Hilbert correspondence is the (zeroth perverse cohomology of the) direct image under the morphism Ψ applied to the intersection complex of Z • .
Finally, we want to state a mirror statement close in spirit to Theorem 5.11 which concerns the reduced quantum D-module. For this, we first need an extension of the localized partial Fourier-Laplace transformation functor FL loc Y as defined in Formula (40) to a functor acting on the category of filtered D-modules. Without giving the actual details (see, e.g. [SY15, Appendix A] or [RS20, Definition 6.2]), let us just state that starting from a filtered D Y -module (M , F • ), this version of the Fourier-Laplace transformation yields an R-module, where again R is the sheaf of Rees rings, as discussed in Section 4.3 (see Formula (28)). We denote this R-module by FL loc C×Y (M , F • ). Moreover, in order to properly state the mirror theorem for nef complete intersections, we have to take into account the so-called mirror map, which was not present in Theorem 5.11 since we restricted our attention to the Fano case there. For a sufficiently small ε ∈ R + , write ∆ * ε := {t ∈ (C * ) r | 0 < |t| < ε} ⊆ K • . Then the mirror map is a morphism Mir : ∆ * ε −→ H 0 (X; C) × U that has been defined in [Giv98,CG07]. Here, U ⊆ K is the set on which the twisted quantum product * tw is defined (converges).
With these preparations, our final mirror theorem can be stated as follows. This result depends in an essential way on the computation of the Hodge filtration on GKZ-systems, that is, on Theorem 4.7, since the expression of the Hodge filtration as the shifted order filtration on the modules M A ′ (β) for various parameters β allows us to describe explicitly the left hand side of (50).
Notice that, by the very definition of the Dubrovin connection, the restriction of the (reduced) quantum D-module to C×∆ * ε has the structure of an R C×∆ * ε -module. A consequence of Theorem 5.17 is the following Hodge theoretic property of the reduced quantum D-module.
The operator Q (2,3) is confluent, univariate and hypergeometric (compare Subsection 1.2) with a regular singularity at q = 0 and irregular singularity at q = ∞.
Notice that if instead we consider a (2, 4)-complete intersection Y ⊂ P 5 , then Y is a Calabi-Yau manifold, and we have 2,4) ), where Q (2,4) = 8q · (2q∂ q + 1)(4q∂ q + 1)(4q∂ q + 2)(4q∂ q + 3) − (q∂ q ) 4 is a homogeneous, hence, regular (non-confluent) hypergeometric operator, with singularities at q = 0, 2 −10 , ∞. In this case, the Hodge theoretic result Corollary 5.18 simply states that D C * q /D C * q · Q (2,4) underlies a pure polarized variation of Hodge structures; this is consistent with [Sim90, Corollary 8.1] and [Del87, Prop. 1.13] (see the discussion on page 33 above). ♦ Finally, let us remark that unlike in the previous example(s), it is in general not easy to give a cyclic description of the intersection cohomology D-module FL loc K • H 0 pr + IC(C T V ×K • ). In other words, even though we know that it has a description as an (Fourier-Laplace transform of an) image of a contiguity morphism, it is not clear how to describe the kernel of this morphism and how to give a presentation of the image as a quotient of D (see also [MM17,Section 6] for some examples and conjectures).

Table of Symbols
Single letters (by alphabet): • A ∈ Z d×n , with columns a 1 , . . . , a n that span ZA = Z d and permit a linear functional having positive values on them.1.5 but also 4.1 for notation in last two sections • B a d × n submatrix of A in final two sections, Convention 4.1 • D 1 , . . . , D n torus invariant divisors on X, Subsection 5.2 • j counts columns (and hence x j , ∂ j , a j ), i counts rows (hence E i ).
• K the complexified Kähler moduli space, the image of H 2 (X; C) under the exponential map, hence the quotient by the integer cohomology lattice scaled by 2π √ −1, Subsection 5.1 • X A affine toric variety and spectrum of S A , closure of T-orbit through (1, . . . , 1), Subsection 2.3 Greek letters and other symbols: