Communications in Number Theory and Physics

Volume 12 (2018)

Number 4

Differential zeros of period integrals and generalized hypergeometric functions

Pages: 609 – 655



Jingyue Chen (Yau Mathematical Sciences Center, Tsinghua University, Beijing China)

An Huang (Department of Mathematics, Brandeis University, Waltham, Massachusetts, U.S.A.)

Bong H. Lian (Department of Mathematics, Brandeis University, Waltham, Massachusetts, U.S.A.)

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)


In this paper, we study the zero loci of locally constant sheaves 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 $V^{\mathsf{v}} = \Gamma (X, K^{-1}{X})$. 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.

Received 17 January 2018

Accepted 14 June 2018

Published 14 January 2019