Alicia Dickenstein, Laura Matusevich, Timur Sadykov
We undertake the study of bivariate Horn systems for generic parameters. We prove that these hypergeometric systems are holonomic, and we provide an explicit formula for their holonomic rank as well as bases of their spaces of complex holomorphic solutions. We also obtain analogous results for the generalized hypergeometric systems arising from lattices of any rank.
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We describe the structure of all codimension-two lattice configurations $A$ which admit a stable rational $A$-hypergeometric function, that is a rational function $F$ all whose partial derivatives are non zero, and which is a solution of the $A$-hypergeometric system of partial differential equations defined by Gel'fand, Kapranov and Zelevinsky. We show, moreover, that all stable rational $A$-hypergeometric functions may be described by toric residues and apply our results to study the rationality of bivariate series whose coefficients are quotients of factorials of linear forms.
We prove that the arithmetic $\mathscr{D}$-modules associated with the $p$-adic generalized hypergeometric differential operators, under a $p$-adic non-Liouvilleness condition on parameters, are described as an iterative multiplicative convolution of (hypergeometric arithmetic) $\mathscr{D}$-modules of rank one. As a corollary, we prove the overholonomicity of hypergeometric arithmetic $\mathscr{D}$-modules under a $p$-adic non-Liouvilleness condition.
We consider mixed Hodge module structures on GKZ-hypergeometric differential systems. We show that the Hodge filtration on these D-modules is given by the order filtration, up to suitable shift. As an application, we prove a conjecture on the existence of non-commutative Hodge structures on the reduced quantum D-module of a nef complete intersection inside a toric variety.
Hypergeometric $\mathcal D$-modules and exponential sums for reductive groups
本文定义了与有限域上reductive群表示家族相关的超几何指数和，引入超几何l-adic层和D-模，证明其是紧致的并估计其秩，利用Wang发展的向量丛一般基上的傅里叶变换理论，展示超几何D-模如何控制超几何层的一般行为，并应用结果估计超几何指数和。
We show that certain one-dimensional hypergeometric differential systems underlie objects of the category of irregular mixed Hodge modules, which was recently introduced by Sabbah, and compute the irregular Hodge filtration for them. We also provide a comparison theorem between two different types of Fourier-Laplace transformation for algebraic integrable twistor $\mathcal{D}$-modules.
The GKZ hypergeometric $\mathcal D$-module
本文研究了GKZ超几何D-模的稳定性，定义了与A-超几何D-模及改进的GKZ系统相关的稳定GKZ超几何D-模，并证明其为单值连通的D-模。
$p$-adic hypergeometric $\mathscr{D}^{\dagger}(\infty)$-module and exponential sums on reductive groups
本文研究了$p$-adic超几何$\mathscr{D}^{\dagger}(\infty)$-模块，用于探讨reductive groups上的指数和，揭示其作为$F$-isocrystals的性质及应用。
Given a family of Laurent polynomials, we will construct a morphism between its (proper) Gauss-Manin system and a direct sum of associated GKZ systems. The kernel and cokernel of this morphism are very simple and consist of free O-modules. The result above enables us to put a mixed Hodge module structure on certain classes of GKZ systems and shows that they have quasi-unipotent monodromy.
A Comparison Theorem for $p$-adic GKZ Hypergeometric $\mathcal D$-Modules and Complexes
本文比较p进GKZ超几何复形与超几何算术D-模，在特定条件下证明比较态射为拟同构，还建立复形的次数集中性、全纯性，构造Frobenius结构并证明其限制为特定过收敛F-等晶体。
The rank of an $A$-hypergeometric $D$-module $M_A(β)$, associated with a full rank $(d\times n)$-matrix $A$ and a vector of parameters $β\in \mathbb{C}^d$, is known to be the normalized volume of $A$, denoted $\mathrm{vol}(A)$, when $β$ lies outside the exceptional arrangement $\mathcal{E}(A)$, an affine subspace arrangement of codimension at least two. If $β\in \mathcal{E}(A)$ is simple, we prove that $d-1$ is a tight upper bound for the ratio $\mathrm{rank}(M_A(β))/\mathrm{vol}(A)$ for any $d\geq 3$. We also prove that the set of parameters $β$ such that this ratio is at least $2$ is an affi…
We describe mirror symmetry for weak toric Fano manifolds as an equivalence of D-modules equipped with certain filtrations. We discuss in particular the logarithmic degeneration behavior at the large radius limit point, and express the mirror correspondence as an isomorphism of Frobenius manifolds with logarithmic poles. The main tool is an identification of the Gauss-Manin system of the mirror Landau-Ginzburg model with a hypergeometric D-module, and a detailed study of a natural filtration defined on this differential system. We obtain a solution of the Birkhoff problem for lattices defined …
Finite Directional Coefficient Modules over Euler Fibers of $A$-Hypergeometric Systems
该研究针对A超几何系统欧拉纤维，在不附加齐次性等条件下，分析定向形式对数解，证明有限个指数格贡献非零解空间，建立Macaulay逆系统对偶关系并给出相关实现准则。
We study the irregularity of hypergeometric D-modules $\mathcal{M}_A (β)$ via the explicit construction of Gevrey series solutions along coordinate subspaces in $X =\mathbb{C}^n$. As a consequence, we prove that along coordinate hyperplanes the combinatorial characterization of the slopes of $\mathcal{M}_A (β)$ given by M. Schulze and U. Walther in [21] still holds without any assumption on the matrix A. We also provide a lower bound for the dimensions of the spaces of Gevrey solutions along coordinate subspaces in terms of volumes of polytopes and prove the equality for very generic parameter…
We study the $p$-adic (generalized) hypergeometric equations by using the theory of multiplicative convolution of arithmetic $\mathscr{D}$-modules. As a result, we prove that the hypergeometric isocrystals with suitable rational parameters have a structure of overconvergent $F$-isocrystals.
Hypergeometric systems from groups with torsion
本文研究了具有torsion的群的超几何系统，证明其为单值系统，并建立了D-模的单值对偶性。
We investigate the GKZ $A$-hypergeometric $\mathscr{D}$-modules, introduced by Gel'fand, Kapranov, and Zelevinskii, arising from cyclic covers of toric varieties and find its Riemann--Hilbert partner. This extends our earlier results in arXiv:1902.01536.
The Euler-Koszul complex is the fundamental tool in the homological study of A-hypergeometric differential systems and functions. We compare Euler-Koszul homology with D-module direct images from the torus to the base space through orbits in the corresponding toric variety. Our approach generalizes a result by Gel'fand et al. and yields a simpler, more algebraic proof. In the process we extend the Euler-Koszul functor a category of infinite toric modules and describe multigraded localizations of Euler-Koszul homology.
Arithmetic hypergeometric $\mathcal {D}$-modules and exponential sums on reductive groups
本文定义约化群有限表示族关联的超几何指数和，引入带Frobenius结构的算术超几何$\boldsymbol{\textrm{D}}$-模研究该指数和，得到其估计并构造相关$F$-等晶体。
We formalize, at the level of D-modules, the notion that A-hypergeometric systems are equivariant versions of the classical hypergeometric equations. For this purpose, we construct a functor on a suitable category of torus equivariant D-modules and show that it preserves key properties, such as holonomicity, regularity, and reducibility of monodromy representation. We also examine its effect on solutions, characteristic varieties, and singular loci. When applied to certain binomial D-modules, our functor produces saturations of the classical hypergeometric differential equations, a fact that s…
For a finite set A of integral vectors, Gel'fand, Kapranov and Zelevinskii defined a system of differential equations with a parameter vector as a D-module, which system is called an A-hypergeometric (or a GKZ hypergeometric) system. Classifying the parameters according to the D-isomorphism classes of their corresponding A-hypergeometric systems is one of the most fundamental problems in the theory. In this paper we give a combinatorial answer for the problem under the assumption that the finite set A lies in a hyperplane off the origin, and illustrate it in two particularly simple cases: the …
Comprehensive Restriction Algorithm for Hypergeometric Systems
针对带参数全纯D-模的限制计算问题，本文给出一般情形的部分解答和超几何情形的完整解答，并基于Oaku的算法提出相应算法。
We compute, by D-module restrictions, the slopes of irregular hypergeometric systems associated to a monomial curve.
The obstruction to intrinsic perturbation for $A$-hypergeometric series
该研究针对A超几何系统的内在扰动问题，构造了内在扰动障碍模，计算了相关余维数，解决了奥山－斋藤留下的开放情形，且在固定格秩下余维数无界。
Let $A$ be a $d$ by $n$ integer matrix. Gel'fand et al. proved that most $A$-hypergeometric systems have an interpretation as a Fourier--Laplace transform of a direct image. The set of parameters for which this happens was later identified by Schulze and Walther as the set of not strongly resonant parameters of $A$. A similar statement relating $A$-hypergeometric systems to exceptional direct images was proved by Reichelt. In this article, we consider a hybrid approach involving neighborhoods $U$ of the torus of $A$ and consider compositions of direct and exceptional direct images. Our main re…
A Dwork family is a one-parameter monomial deformation of a Fermat hypersurface. In this paper we compute algebraically the invariant part of its Gauss-Manin cohomology under the action of certain subgroup of automorphisms. To achieve that goal we use the algebraic theory of $\mathcal{D}$-modules, especially one-dimensional hypergeometric ones.
We study the mixed twistor D-modules associated to meromorphic functions. In particular, we describe their push-forward and specialization under some situations. We apply the results to study the twistor property of a type of GKZ-hypergeometric systems, and to study their specializations. As a result, we obtain some isomorphisms of mixed TEP-structures in the local mirror symmetry.
Hypergeometric integrals, hook formulas and Whittaker vectors
本文确定了两类与Young图相关的多维超几何积分之间的比例系数，即加权钩子乘积的倒数，并由此给出gl_n泛包络代数中心在Whittaker向量空间上作用的显式同时特征向量基。
The Gamma-series of Gel'fand-Kapranov-Zelevinsky are adapted so that they give solutions for certain resonant systems of GKZ hypergeometric differential equations. For this some complex parameters in the Gamma-series are replaced by nilpotent elements from a ring $R_{A,T}$. The adapted Gamma-series is a function $Ψ$ with values in the finite dimensional vector space $R_{A,T}\otimes C$. Applications of these results in the context of toric Mirror Symmetry are described. Building on work of Batyrev we show that the relative cohomology module of a certain hypersurface in a torus is a GKZ hypergeo…
We study $A$-hypergeometric systems $H_A(β)$ in the sense of Gelfand, Kapranov and Zelevinsky under two aspects: the structure of their holonomically dual system, and reducibility of their rank module. We prove first that rank-jumping parameters always correspond to reducible systems, and we show that the property of being reducible is ``invariant modulo the lattice''. In the second part we study a conjecture of Nobuki Takayama which states that the holonomic dual of $H_A(β)$ is of the form $H_A(β')$ for suitable $β'$. We prove the conjecture for all matrices $A$ and generic parameter $β$, exh…
The holonomic rank of the A-hypergeometric system M_A(β) is the degree of the toric ideal I_A for generic parameters; in general, this is only a lower bound. To the semigroup ring of A we attach the ranking arrangement and use this algebraic invariant and the exceptional arrangement of nongeneric parameters to construct a combinatorial formula for the rank jump of M_A(β). As consequences, we obtain a refinement of the stratification of the exceptional arrangement by the rank of M_A(β) and show that the Zariski closure of each of its strata is a union of translates of linear subspaces of the pa…
Characteristic cycles and the Gevrey filtration of irregularity in hypergeometric families
该研究证明了A-超几何系统特征标圈重数的上半连续性，证实了Schulze与Walther的猜想，还构造了特定半群环揭示Gevrey跳跃轨迹的性质。
We study the irregularity sheaves attached to the $A$-hypergeometric $D$-module $M_A(β)$ introduced by Gel'fand et al., where $A\in\mathbb{Z}^{d\times n}$ is pointed of full rank and $β\in\mathbb{C}^d$. More precisely, we investigate the slopes of this module along coordinate subspaces. In the process we describe the associated graded ring to a positive semigroup ring for a filtration defined by an arbitrary weight vector $L$ on torus equivariant generators. To this end we introduce the $(A,L)$-umbrella, a simplicial complex determined by $A$ and $L$, and identify its facets with the component…
Tautological systems, homogeneous spaces and the holonomic rank problem
本文通过函子性构造，将混合Hodge模块结构推广到齐性空间相关的全息系统，并解决了其全消元秩问题。
We study quotients of the Weyl algebra by left ideals whose generators consist of an arbitrary Z^d-graded binomial ideal I along with Euler operators defined by the grading and a parameter in C^d. We determine the parameters for which these D-modules (i) are holonomic (equivalently, regular holonomic, when I is standard-graded); (ii) decompose as direct sums indexed by the primary components of I; and (iii) have holonomic rank greater than the generic rank. In each of these three cases, the parameters in question are precisely those outside of a certain explicitly described affine subspace arr…
We study binomial D-modules, which generalize A-hypergeometric systems. We determine explicitly their singular loci and provide three characterizations of their holonomicity. The first of these states that a binomial D-module is holonomic if and only if its corresponding singular locus is proper. The second characterization is an equivalence of holonomicity and L-holonomicity for these systems. The third refines the second by giving more detailed information about the L-characteristic variety of a non-holonomic binomial D-module.
We prove that a holonomic binomial $D$--module $M_A (I,β)$ is regular if and only if certain associated primes of $I$ determined by the parameter vector $β\in \CC^d$ are homogeneous. We further describe the slopes of $M_A(I,β)$ along a coordinate subspace in terms of the known slopes of some related hypergeometric $D$--modules that also depend on $β$. When the parameter $β$ is generic, we also compute the dimension of the generic stalk of the irregularity of $M_A(I,β)$ along a coordinate hyperplane and provide some remarks about the construction of its Gevrey solutions.
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