Sections of Hodge bundles I: Global theory and applications to period maps

TL;DR

This paper investigates the global properties of Hodge bundles and period maps, providing new geometric insights and explicit realizations, and applies these results to construct a complex affine structure on Teichmüller space.

Contribution

It introduces a novel combination of deformation theory and matrix representations to analyze period maps, partially confirming Griffiths' conjecture and constructing a global affine structure.

Findings

The image of the lifted period map is contained in a complex Euclidean subspace.

A global complex affine structure is constructed on Teichmüller space of Calabi--Yau type manifolds.

Partial verification of Griffiths' conjecture on period maps.

Abstract

We study global sections of Hodge bundles arising from two complementary constructions: a deformation-theoretic construction, which yields global geometric consequences for period maps, and a construction from the matrix representation of the image of the period map, which provides an explicit Euclidean realization. Combining these perspectives, we prove that the image of the lifted period map on the universal cover is contained in a complex Euclidean subspace of the period domain, thereby giving a partial solution to a conjecture of Griffiths on the global behavior of period maps. As an application, we construct a global complex affine structure on the Teichm\"uller space of Calabi--Yau type manifolds.