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A GKZ D-module, introduced by Gelfand, Kapranov, and Zelevinsky, is a system of linear PDEs generalizing the hypergeometric structure that can be traced back to Euler and Gauss. GKZ D-modules are powerful tools in the study of the moduli space of Calabi–Yau manifolds as they govern the period integrals for Calabi–Yau complete intersections/double covers in toric varieties. Thus, to compute period integrals, one could compute the full solution space to such a GKZ D-module first, and then try to recover the periods among the solutions.
In this talk, I will mainly focus on the case of elliptic curves and give a cohomological description of the solution space to such a GKZ D-module. If time permits, I will also talk about the higher dimensional case. This is based on joint works with Dingxin Zhang.