Period mappings and properties of the augmented Hodge line bundle

TL;DR

This paper explores the structure and compactification of the image of period maps in Hodge theory, proposing a conjectural completion with complex analytic and algebraic properties, verified in low dimensions.

Contribution

It introduces a conjectural Hodge theoretic completion of period map images and proves its structure as a projective variety in low dimensions, using positivity properties of Hodge bundles.

Findings

The set P is given a compact Hausdorff topology.

The conjecture that P admits a complex analytic structure is verified when dim P   2.

The augmented Hodge line bundle extends to an ample line bundle on the completion, making it a projective variety.

Abstract

Let $P$ be the image of a period map. We discuss progress towards a conjectural Hodge theoretic completion $\overline{P}$, an analogue of the Satake-Baily-Borel compactification in the classical case. The set $\overline{P}$ is defined and given the structure of a compact Hausdorff topological space. We conjecture that it admits the structure of a compact complex analytic variety. We verify this conjecture when $\mathrm{dim} P \le 2$. In general, $\overline{P}$ admits a finite cover $\overline{S}$ (also a compact Hausdorff space, and constructed from Stein factorizations of period maps). Assuming that $\overline{S}$ is a compact complex analytic variety, we show that a lift of the augmented Hodge line bundle $\Lambda$ extends to an ample line bundle, giving $\overline{P}$ the structure of a projective normal variety. Our arguments rely on refined positivity properties of Chern forms associated to various Hodge bundles; properties that might be of independent interest.