On the collapsing of Calabi-Yau manifolds and K\"ahler-Ricci flows

TL;DR

This paper investigates the collapsing behavior of Calabi-Yau metrics and Kähler-Ricci flows on fiber spaces with smooth bases, providing explicit bounds on singular sets and conditions for metric limits.

Contribution

It identifies the Gromov-Hausdorff limits of Kähler-Ricci flows under specific geometric conditions and relates metric behavior to birational geometry.

Findings

Explicit bounds for the Hausdorff measure of singular sets

Identification of Gromov-Hausdorff limits in collapsing scenarios

Conditions linking birational geometry to metric limits

Abstract

We study the collapsing of Calabi-Yau metrics and of Kahler-Ricci flows on fiber spaces where the base is smooth. We identify the collapsed Gromov-Hausdorff limit of the Kahler-Ricci flow when the divisorial part of the discriminant locus has simple normal crossings. In either setting, we also obtain an explicit bound for the real codimension 2 Hausdorff measure of the Cheeger-Colding singular set, and identify a sufficient condition from birational geometry to understand the metric behavior of the limiting metric on the base.