GKZ-generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces
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.
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