Degenerations of Ricci-flat Calabi-Yau manifolds

TL;DR

This paper studies how Ricci-flat Kähler metrics on Calabi-Yau manifolds behave under degenerations, extending previous results by removing the need for crepant resolutions, thus broadening understanding of metric limits in complex geometry.

Contribution

It generalizes earlier convergence results by eliminating the crepant resolution condition, advancing the theory of metric degenerations of Calabi-Yau manifolds.

Findings

Established Gromov-Hausdorff convergence without crepant resolution assumptions

Extended previous theorems to a broader class of Calabi-Yau degenerations

Provided new insights into the metric limits of degenerating Ricci-flat Kähler manifolds

Abstract

This paper is a sequel to arXiv:1012.2940. We further investigate the Gromov-Hausdorff convergence of Ricci-flat K\"{a}hler metrics under degenerations of Calabi-Yau manifolds. We extend Theorem 1.1 in arXiv:1012.2940 by removing the condition on existence of crepant resolutions for Calabi-Yau varieties.