Infinite time singularities of the K\"ahler-Ricci flow

TL;DR

This paper analyzes the long-term behavior of the Kähler-Ricci flow on compact Kähler manifolds, classifying singularities at infinity and describing convergence properties in fibered cases with semiample canonical bundles.

Contribution

It provides a near-complete classification of the flow's singularity types at infinity based solely on the complex structure, and describes convergence to product metrics in fibered cases.

Findings

Flow singularities depend only on the complex structure.

Rescalings around points on smooth fibers converge to product Ricci-flat and flat metrics.

Results extend to collapsing limits of Ricci-flat Kähler metrics.

Abstract

We study the long-time behavior of the Kahler-Ricci flow on compact Kahler manifolds. We give an almost complete classification of the singularity type of the flow at infinity, depending only on the underlying complex structure. If the manifold is of intermediate Kodaira dimension and has semiample canonical bundle, so that it is fibered by Calabi-Yau varieties, we show that parabolic rescalings around any point on a smooth fiber converge smoothly to a unique limit, which is the product of a Ricci-flat metric on the fiber and of a flat metric on Euclidean space. An analogous result holds for collapsing limits of Ricci-flat Kahler metrics.