On the better behaved version of the GKZ hypergeometric system
@article{Borisov2010OnTB, title={On the better behaved version of the GKZ hypergeometric system}, author={Lev Borisov and R. Paul Horja}, journal={Mathematische Annalen}, year={2010}, volume={357}, pages={585-603} }
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…
14 Citations
Toric Deligne-Mumford stacks and the better behaved version of the GKZ hypergeometric system
- Mathematics
- 2012
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
- Mathematics
- 2020
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
- MathematicsAsian Journal of Mathematics
- 2021
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
- Mathematics
- 2014
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
- Mathematics
- 2011
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
- MathematicsMoscow 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
- MathematicsSymmetry, 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
- Mathematics
- 2022
. 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
- Mathematics
- 2012
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
- Mathematics
- 2016
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…
References
SHOWING 1-10 OF 16 REFERENCES
Rational Hypergeometric Functions
- MathematicsCompositio Mathematica
- 2001
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
- Mathematics
- 1996
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
- Mathematics
- 1997
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
- Mathematics
- 2004
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
- Mathematics
- 1993
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
- Mathematics
- 2000
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
- Mathematics
- 2011
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
- Mathematics
- 2004
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⋆
- MathematicsCompositio Mathematica
- 2003
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…