Resonant Hypergeometric Systems and Mirror Symmetry

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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 Rₐ,ₜ. The adapted Gamma-series is a function Ψ with values in the finite dimensional vector space Rₐ,ₜ⊗ 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 hypergeometric D-module which over an appropriate domain is isomorphic to the trivial D-module Rₐ,ₜ⊗ Oₜ, where Oₜ is the sheaf of holomorphic functions on this domain. The isomorphism is explicitly given by adapted Gamma-series. As a result one finds the periods of a holomorphic differential form of degree d on a d-dimensional Calabi-Yau manifold, needed for the B-model side input to Mirror Symmetry. Relating our work with that of Batyrev and Borisov we interpret the ring, as the cohomology ring of a toric variety and a certain principal ideal in it as a subring of the Chow ring of a Calabi-Yau complete intersection. This interpretation takes place on the A-model side of Mirror Symmetry.