Ricci flow on quasiprojective manifolds

TL;DR

This paper studies the behavior of the Kähler-Ricci flow on non-compact, finite-volume manifolds derived from compact Kähler manifolds, providing formulas for singularity timing and conditions for type-II singularities.

Contribution

It introduces a method to compute singularity time from cohomological data and establishes conditions for type-II singularities in this setting.

Findings

Singularity time expressed via cohomological data

Sufficient condition for type-II singularities

Analysis of asymptotic behavior of the flow

Abstract

We consider the Kaehler-Ricci flow on complete finite-volume metrics that live on the complement of a divisor in a compact Kaehler manifold X. Assuming certain spatial asymptotics on the initial metric, we compute the singularity time in terms of cohomological data on X. We also give a sufficient condition for the singularity, if there is one, to be type-II.