Completion of the moduli space for polarized Calabi-Yau manifolds

TL;DR

This paper constructs a completion of the moduli space for polarized Calabi-Yau manifolds using Ricci-flat metrics and Gromov-Hausdorff topology, analyzing its geometric and algebraic properties.

Contribution

It introduces a new method to complete the moduli space of polarized Calabi-Yau manifolds, incorporating singular varieties and studying their geometric structure.

Findings

The completion can be exhausted by sequences of quasi-projective varieties.

New points added have finite Weil-Petersson distance to the interior.

The study reveals algebro-geometric and Weil-Petersson geometric properties of the completion.

Abstract

In this paper, we construct a completion of the moduli space for polarized Calabi-Yau manifolds by using Ricci-flat K\"ahler-Einstein metrics and the Gromov-Hausdorff topology, which parameterizes certain Calabi-Yau varieties. We then study the algebro-geometric perperties and the Weil-Petersson geometry of such completion. We show that the completion can be exhausted by sequences of quasi-projective varieties, and new points added have finite Weil-Petersson distance to the interior.