Fano Fibrations and Twisted K\"ahler-Einstein Metrics II: The K\"ahler-Ricci Flow

TL;DR

This paper analyzes the behavior of the K"ahler-Ricci flow on Fano fibrations, establishing diameter bounds, potential estimates, and scalar curvature behavior near singularities, advancing understanding of geometric collapse in complex geometry.

Contribution

It provides new diameter bounds, potential estimates, and curvature results for the K"ahler-Ricci flow on Fano fibrations with singularities, extending previous work to include detailed collapse rates.

Findings

Diameter of fibers collapses at rate √(T-t)

Potential estimates for complex Monge-Ampère flow

Type I scalar curvature behavior away from singular fibers

Abstract

This is the second of two papers studying both the geometric structure of Fano fibrations and the application to K\"ahler-Ricci flows developing a singularity in finite time. We assume that the K\"ahler-Ricci flow on a compact K\"ahler manifold has a rational initial metric and develops a singularity in finite time such that the manifold admits a Fano fibration structure. Moreover, it is assumed that the volume form of the flow collapses uniformly at the rate of $C^{-1}(T-t)^{n-m} \Omega \leq \omega(t)^n\leq C(T-t)^{n-m}\Omega$. Under this setting, a diameter bound is obtained in any compact set away from singular fibres and the diameter of the fibres is proven to collapse at the optimal rate $\sqrt{T-t}$. Furthermore, several precise $C^0$-estimates are proven for the potential of the complex Monge-Ampere flow which involve the potentials of singular twisted K\"ahler-Einstein metrics on the base variety from part I. Finally, in the case of K\"ahler-Einstein Fano fibres, we deduce Type I scalar curvature in any compact set away from singular fibres and globally for a submersion.