The K\"ahler-Ricci flow, Ricci-flat metrics and collapsing limits

TL;DR

This paper studies the behavior of the Kähler-Ricci flow on fiber spaces with Calabi-Yau fibers, showing convergence to base metrics and Ricci-flat metrics on fibers, extending previous results.

Contribution

It provides new uniform convergence results for the Kähler-Ricci flow on fiber spaces and analyzes degenerations of Ricci-flat Kähler metrics.

Findings

Uniform metric convergence to the base away from singular fibers

Rescaled fiber metrics converge to Ricci-flat Kähler metrics

Results extend previous work by Song-Tian and others

Abstract

We investigate the Kahler-Ricci flow on holomorphic fiber spaces whose generic fiber is a Calabi-Yau manifold. We establish uniform metric convergence to a metric on the base, away from the singular fibers, and show that the rescaled metrics on the fibers converge to Ricci-flat Kahler metrics. This strengthens previous work of Song-Tian and others. We obtain analogous results for degenerations of Ricci-flat Kahler metrics.