On the collapsing rate of the K\"ahler-Ricci flow with finite-time singularity

TL;DR

This paper investigates the collapsing behavior of the K"ahler-Ricci flow on certain compact K"ahler manifolds, establishing conditions for optimal fiber collapse rate near finite-time singularities.

Contribution

It provides cohomological and curvature criteria that determine the precise collapse rate of fibers during the flow's finite-time singularity.

Findings

Fibers collapse at the rate ~(T-t)^{1/2} under specified conditions.

Cohomological and curvature conditions influence collapse behavior.

Results clarify the geometry of K"ahler-Ricci flow near singularities.

Abstract

This short note studies the collapsing behavior of the K\"ahler-Ricci flow on a compact K\"ahler manifold X admitting a holomorphic submersion X -> B where B is a K\"ahler manifold of lower dimension than X. We give cohomological and curvature conditions under which the fibers collapse at the optimal rate ~(T-t)^{1/2}