Period integrals and Hodge modules

TL;DR

This paper introduces a new map associated with polarized Hodge modules that generalizes the classical period integral pairing, connecting Hodge theory with Hodge modules and their extensions.

Contribution

It defines a novel map $al{P}_M$ for polarized Hodge modules and analyzes its properties, extending classical period integrals to the setting of Hodge modules.

Findings

The map $al{P}_M$ generalizes period pairings to Hodge modules.

In the case of minimal extensions, the homotopy image matches the minimal extension of the period map.

The construction bridges classical Hodge theory and the theory of Hodge modules.

Abstract

We define a map $\mathcal{P}_M$ attached to any polarized Hodge module $M$ such that the restriction of $\mathcal{P}_M$ to a locus on which $M$ is a variation of Hodge structures induces the usual period integral pairing for this variation of Hodge structures. In the case that $M$ is the minimal extension of a simple polarized variation of Hodge structures $V$, we show that the homotopy image of $\mathcal{P}_M$ is the minimal extension of the graph morphism of the usual period integral map for $V$.