A Global Torelli Theorem for Calabi-Yau Manifolds

TL;DR

This paper proves that the period map for Calabi-Yau manifolds is an embedding, using holomorphic affine structures and Hodge metric completions, advancing the understanding of their moduli spaces.

Contribution

It establishes a global Torelli theorem for Calabi-Yau manifolds by showing the period map is an embedding, with new constructions of affine structures and a canonical section.

Findings

Period map is an embedding for Calabi-Yau manifolds.

Constructs a holomorphic affine structure on Teichmüller space.

Provides a canonical global holomorphic section of the (n,0) class.

Abstract

We describe the proof that the period map from the Torelli space of Calabi-Yau manifolds to the classifying space of polarized Hodge structures is an embedding. The proof is based on the constructions of holomorphic affine structure on the Teichm\"uller space and Hodge metric completion of the Torelli space. A canonical global holomorphic section of the holomorphic $(n, 0)$ class on the Teichm\"uller space is constructed.