On the better behaved version of the GKZ hypergeometric system

  title={On the better behaved version of the GKZ hypergeometric system},
  author={Lev Borisov and R. Paul Horja},
  journal={Mathematische Annalen},
We consider a version of the generalized hypergeometric system introduced by Gelfand, Kapranov and Zelevinsky (GKZ) suited for the case when the underlying lattice is replaced by a finitely generated abelian group. In contrast to the usual GKZ hypergeometric system, the rank of the better behaved GKZ hypergeometric system is always the expected one. We give largely self-contained proofs of many properties of this system. The discussion is intimately related to the study of the variations of… 
Toric Deligne-Mumford stacks and the better behaved version of the GKZ hypergeometric system
We generalize the combinatorial description of the orbifold (Chen--Ruan) cohomology and of the Grothendieck ring of a Deligne--Mumford toric stack and its associated stacky fan in a lattice $N$ in
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
On duality of certain GKZ hypergeometric systems
We study a pair of conjectures on better behaved GKZ hypergeometric systems of PDEs inspired by Homological mirror symmetry for crepant resolutions of Gorenstein toric singularities. We prove the
On stringy cohomology spaces
We modify the definition of the families of $A$ and $B$ stringy cohomology spaces associated to a pair of dual reflexive Gorenstein cones. The new spaces have the same dimension as the ones defined
Quantum Cohomology and Periods
In a previous paper, the author introduced a Z-structure in quantum cohomology defined by the K-theory and the Gamma class and showed that it is compatible with mirror symmetry for toric orbifolds.
Period Integrals Associated to an Affine Delsarte Type Hypersurface
  • S. Tanabé
  • Mathematics
    Moscow Mathematical Journal
  • 2022
We calculate the period integrals for a special class of affine hypersurfaces (deformed Delsarte hypersurfaces) in an algebraic torus by the aid of their Mellin transforms. A description of the
Global Mirrors and Discrepant Transformations for Toric Deligne-Mumford Stacks
  • H. Iritani
  • Mathematics
    Symmetry, Integrability and Geometry: Methods and Applications
  • 2020
We introduce a global Landau-Ginzburg model which is mirror to several toric Deligne-Mumford stacks and describe the change of the Gromov-Witten theories under discrepant transformations. We prove a
Discriminants and toric K-theory
. We discuss a categorical approach to the theory of discriminants in the combinatorial language introduced by Gelfand, Kapranov and Zelevinsky. Our point of view is inspired by homological mirror
Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and Orlov equivalence
We show that the Gromov-Witten theory of Calabi-Yau hypersurfaces matches, in genus zero and after an analytic continuation, the quantum singularity theory (FJRW theory) recently introduced by Fan,
Hodge-theoretic mirror symmetry for toric stacks
Using the mirror theorem [CCIT15], we give a Landau-Ginzburg mirror description for the big equivariant quantum cohomology of toric Deligne-Mumford stacks. More precisely, we prove that the big


Rational Hypergeometric Functions
Multivariate hypergeometric functions associated with toric varieties were introduced by Gel'fand, Kapranov and Zelevinsky. Singularities of such functions are discriminants, that is, divisors
Maximal Degeneracy Points of GKZ Systems
Motivated by mirror symmetry, we study certain integral representations of solutions to the Gel'fand-Kapranov-Zelevinsky(GKZ) hypergeometric system. Some of these solutions arise as period integrals
Resonant Hypergeometric Systems and Mirror Symmetry
In Part I the Γ-series of [11] are adapted so that they give solutions for certain resonant systems of Gel’fand-Kapranov-Zelevinsky hypergeometric differential equations. For this some complex
Homological methods for hypergeometric families
We analyze the behavior of the holonomic rank in families of holonomic systems over complex algebraic varieties by providing homological criteria for rank-jumps in this general setting. Then we
Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties
We formulate general conjectures about the relationship between the A-model connection on the cohomology of ad-dimensional Calabi-Yau complete intersectionV ofr hypersurfacesV1,...,Vr in a toric
Grbner Deformations of Hypergeometric Differential Equations
The theory of Grbner bases is a main tool for dealing with rings of differential operators. This book reexamines the concept of Grbner bases from the point of view of geometric deformations. The
Quantum Cohomology and Periods
In a previous paper, the author introduced a Z-structure in quantum cohomology defined by the K-theory and the Gamma class and showed that it is compatible with mirror symmetry for toric orbifolds.
The orbifold Chow ring of toric Deligne-Mumford stacks
Generalizing toric varieties, we introduce toric Deligne-Mumford stacks which correspond to combinatorial data. The main result in this paper is an explicit calculation of the orbifold Chow ring of a
Intersection Cohomology on Nonrational Polytopes⋆
AbstractWe consider a fan as a ringed space (with finitely many points). We develop the corresponding sheaf theory and functors, such as direct image Rπ* (π is a subdivision of a fan), Verdier