Simultaneous normalization of period map and affine structures on moduli spaces

TL;DR

This paper demonstrates that the lifted period map's image resides in a complex Euclidean space and establishes that certain Teichmüller spaces possess complex affine structures, advancing understanding of moduli space geometry.

Contribution

It proves the image of the lifted period map is in a complex Euclidean space and shows specific Teichmüller spaces have complex affine structures, providing new insights into moduli space geometry.

Findings

Lifted period map image lies in complex Euclidean space

Teichmüller spaces of certain polarized manifolds have complex affine structures

Advances understanding of moduli space geometry

Abstract

We prove that the image of the lifted period map on the universal cover lies in a complex Euclidean space. We also prove that the Teichm\"uller spaces of a class of polarized manifolds have complex affine structures.