Limits of Calabi-Yau metrics when the Kahler class degenerates

TL;DR

This paper investigates how Ricci-flat Kähler metrics on Calabi-Yau manifolds behave as their Kähler classes degenerate, showing convergence to incomplete Ricci-flat metrics outside a subvariety when the limit class is big and nef.

Contribution

It provides a detailed analysis of metric degeneration in Calabi-Yau manifolds and connects geometric limits with algebraic geometry insights.

Findings

Ricci-flat metrics converge smoothly outside a subvariety

Limit metrics are incomplete Ricci-flat when the class is big and nef

Algebraic geometry describes the limit behavior

Abstract

We study the behaviour of families of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold when the Kahler classes degenerate to the boundary of the ample cone. We prove that if the limit class is big and nef the Ricci-flat metrics converge smoothly on compact sets outside a subvariety to a limit incomplete Ricci-flat metric. The limit can also be understood from algebraic geometry.