Finite time collapsing of the K\"ahler-Ricci flow on threefolds

TL;DR

This paper investigates the behavior of the Kähler-Ricci flow on compact threefolds, showing that finite-time singularities imply the existence of a Fano fibration, and finite-time extinction characterizes Fano manifolds.

Contribution

It establishes a link between finite-time singularities in the Kähler-Ricci flow and the Fano fibration structure of threefolds, providing new geometric insights.

Findings

Finite-time collapsing singularities imply a Fano fibration.

Finite-time extinction occurs if and only if the manifold is Fano.

Initial class is proportional to the first Chern class in extinction case.

Abstract

We show that if on a compact Kahler threefold there is a solution of the Kahler-Ricci flow which encounters a finite time collapsing singularity, then the manifold admits a Fano fibration. Furthermore, if there is finite time extinction then the manifold is Fano and the initial class is a positive multiple of the first Chern class.