Collapsing immortal K\"ahler-Ricci flows

TL;DR

This paper studies the behavior of the K"ahler-Ricci flow on certain compact K"ahler manifolds, showing it collapses to a canonical metric on the base of the fibration, confirming two conjectures.

Contribution

It proves that the K"ahler-Ricci flow collapses to a canonical metric on the base of the Iitaka fibration for manifolds with semiample canonical bundle and intermediate Kodaira dimension, confirming conjectures.

Findings

Flow collapses to a canonical metric on the base

Ricci curvature remains bounded away from singular fibers

Provides asymptotic expansion for evolving metrics

Abstract

We consider the K\"ahler-Ricci flow on compact K\"ahler manifolds with semiample canonical bundle and intermediate Kodaira dimension, and show that the flow collapses to a canonical metric on the base of the Iitaka fibration in the locally smooth topology and with bounded Ricci curvature away from the singular fibers. This follows from an asymptotic expansion for the evolving metrics, in the spirit of recent work of the first and third-named authors on collapsing Calabi-Yau metrics, and proves two conjectures of Song and Tian.