Geometry of twisted K\"ahler-Einstein metrics and collapsing

TL;DR

This paper investigates the geometry of twisted Kähler-Einstein metrics on the base of certain fiber spaces, revealing their singularities and behavior during collapsing processes in Calabi-Yau manifolds.

Contribution

It establishes the presence of conical singularities in twisted Kähler-Einstein metrics and analyzes their limits during collapsing, advancing understanding of metric degenerations.

Findings

Twisted Kähler-Einstein metrics have conical singularities along the discriminant locus.

The paper describes the Gromov-Hausdorff limits of collapsing Calabi-Yau metrics.

Results apply to the study of Kähler-Ricci flow and metric degenerations.

Abstract

We prove that the twisted Kahler-Einstein metrics that arise on the base of certain holomorphic fiber space with Calabi-Yau fibers have conical-type singularities along the discriminant locus. These fiber spaces arise naturally when studying the collapsing of Ricci-flat Kahler metrics on Calabi-Yau manifolds, and of the Kahler-Ricci flow on compact Kahler manifolds with semiample canonical bundle and intermediate Kodaira dimension. Our results allow us to understand their collapsed Gromov-Hausdorff limits when the base is smooth and the discriminant has simple normal crossings.