On a conjecture of Candelas and de la Ossa

TL;DR

This paper proves that the metric completion of a canonical Ricci-flat Kähler metric on a Calabi-Yau variety with singularities is homeomorphic to the variety, confirming a conjecture related to conifold transitions.

Contribution

It establishes the homeomorphism of the metric completion to the original variety and confirms a conjecture of Candelas and de la Ossa for conifold flops and transitions.

Findings

The metric completion is homeomorphic to the original variety.

Confirmed the conjecture for conifold flops.

Provided a framework for understanding Ricci-flat metrics on singular Calabi-Yau varieties.

Abstract

We prove that the metric completion of a canonical Ricci-flat Kahler metric on the nonsingular part of a projective Calabi-Yau variety $X$ with ordinary double point singularities, is a compact metric length space homeomorphic to the projective variety $X$ itself. As an application, we prove a conjecture of Candelas and de la Ossa for conifold flops and transitions.