The Collapsing Rate of the K\"ahler-Ricci Flow with Regular Infinite Time Singularity

TL;DR

This paper analyzes the collapsing behavior of the Kähler-Ricci flow on certain compact Kähler manifolds, showing the flow's metric degenerates at an exponential rate, with implications for geometric extensions and Calabi-Yau fibrations.

Contribution

It demonstrates the precise collapsing rate of the Kähler-Ricci flow on manifolds with holomorphic submersions, extending previous work to regular and possibly singular Calabi-Yau fibrations.

Findings

Flow metric degenerates at rate e^{-t}

Fibers collapse at diameter ~ e^{-t/2}

Extensions to regular and singular fibrations

Abstract

We study the collapsing behavior of the Kaehler-Ricci flow on a compact Kaehler manifold X admitting a holomorphic submersion X -> S coming from its canonical class, where S is a Kaehler manifold with dim S < dim X. We show that the flow metric degenerates at exactly the rate of e^{-t} as predicted by the cohomology information, and so the fibers collapse at the optimal rate diameter ~ e^{-t/2}. Consequently, it leads to some analytic and geometric extensions to the regular case of Song-Tian's works on elliptic and Calabi-Yau fibrations. Its applicability to general Calabi-Yau fibrations with possibly singular fibers will also be discussed in local sense.