Period Integrals of CY and General Type Complete Intersections

TL;DR

This paper develops a global residue formula to analyze period integrals of CY and general type complete intersections, providing explicit systems like Picard-Fuchs equations in various geometric contexts.

Contribution

It introduces a new global Poincaré residue formula for period integrals of CY and general type complete intersections, extending previous methods and incorporating automorphism groups and tautological systems.

Findings

Explicit Picard-Fuchs systems for complete intersections in Fano varieties

Construction of a holonomic D-module governing period integrals

Application to CY and general type varieties in homogeneous spaces

Abstract

We develop a global Poincar\'e residue formula to study period integrals of families of complex manifolds. For any compact complex manifold $X$ equipped with a linear system $V^*$ of generically smooth CY hypersurfaces, the formula expresses period integrals in terms of a canonical global meromorphic top form on $X$. Two important ingredients of our construction are the notion of a CY principal bundle, and a classification of such rank one bundles. We also generalize our construction to CY and general type complete intersections. When $X$ is an algebraic manifold having a sufficiently large automorphism group $G$ and $V^*$ is a linear representation of $G$, we construct a holonomic D-module that governs the period integrals. The construction is based in part on the theory of tautological systems we have developed in the paper \cite{LSY1}, joint with R. Song. The approach allows us to explicitly describe a Picard-Fuchs type system for complete intersection varieties of general types, as well as CY, in any Fano variety, and in a homogeneous space in particular. In addition, the approach provides a new perspective of old examples such as CY complete intersections in a toric variety or partial flag variety.