GKZ-generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces
type: Item
instance of: scholarly article
title: GKZ-generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces
zbMATH Open document ID: 0870.14028
DOI: 10.1007/BF02506417
author: Bong H. Lian
author: Shinobu Hosono
author: Shing-Tung Yau
published in: Communications in Mathematical Physics
publication date: 1997-09-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/alg-geom/9511001
review text: It is known that the Picard-Fuchs equations for the periods of Calabi-Yau hypersurfaces in a toric Fano variety \({\mathbb{P}}_{\Delta}\) coincide with a generalized hypergeometric system introduced by Gelfand-Kapranov-Zelevinski [see \textit{V. V. Batyrev} and \textit{D. van Straaten}, Commun. Math. Phys. 168, No. 3, 493-533 (1995; Zbl 0843.14016)]. Similarly, the periods of the mirror family in \({\mathbb{P}}_{\Delta^*}\) satisfy a GKZ-hypergeometric system. Although this system is usually reducible, its irreducible part can be extracted by introducing an extended system which incorporates additional differential operators corresponding to the automorphism group of \({\mathbb{P}}_{\Delta^*}\). The authors show that the local properties of this system are determined by an algebro-combinatorial object, known as a toric ideal. Its Gröbner basis determines a finite set of differential operators for the local solutions of the GKZ-system. In the case of 3-dimensional varieties with \(h^{11}\leq 3\) they give some more explicit analysis of local solutions as well as some examples.
Mathematics Subject Classification ID: 14J32
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 14D05
Mathematics Subject Classification ID: 14J70
Mathematics Subject Classification ID: 81T30
Mathematics Subject Classification ID: 33C50
zbMATH DE Number: 975294
zbMATH Keywords: Picard-Fuchs equations
zbMATH Keywords: Calabi-Yau hypersurfaces
zbMATH Keywords: periods
zbMATH Keywords: mirror family
zbMATH Keywords: toric ideal
zbMATH Keywords: Gröbner basis
zbMATH Keywords: toric Fano variety
reviewed by: Igor V. Dolgachev
MaRDI profile type: Publication
arXiv ID: alg-geom/9511001
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