Differential zeros of period integrals and generalized hypergeometric functions

TL;DR

This paper investigates the algebraic structure and explicit equations of the zero loci of period integrals associated with Calabi-Yau hypersurfaces, revealing their stratification and non-emptiness in certain cases.

Contribution

It provides new descriptions and computational methods for the zero loci of period integrals, extending previous results and establishing their algebraic nature.

Findings

Zero loci are algebraic and sometimes non-empty.

Explicit polynomial equations for zero loci are derived in specific cases.

A natural stratification of the zero locus is introduced.

Abstract

In this paper, we study the zero loci of local systems of the form $\delta\Pi$, where $\Pi$ is the period sheaf of the universal family of CY hypersurfaces in a suitable ambient space $X$, and $\delta$ is a given differential operator on the space of sections $V^\vee=\Gamma(X,K_X^{-1})$. Using earlier results of three of the authors and their collaborators, we give several different descriptions of the zero locus of $\delta\Pi$. As applications, we prove that the locus is algebraic and in some cases, non-empty. We also give an explicit way to compute the polynomial defining equations of the locus in some cases. This description gives rise to a natural stratification to the zero locus.