Expanding Soliton Models for K\"ahler-Ricci Flow Near Conical Singularities

TL;DR

This paper studies the behavior of the K"ahler-Ricci flow on singular spaces with cone-like singularities, showing curvature bounds and asymptotic models near singular points.

Contribution

It extends soliton models for K"ahler-Ricci flow to spaces with conical singularities, providing geometric descriptions near these points.

Findings

K"ahler-Ricci flow satisfies a $C/t$ curvature bound near singularities.

Flow near each singular point is modeled on a unique K"ahler-Ricci expander.

Results aid understanding of singularities in the minimal model program.

Abstract

Let $(Y,g_0)$ be a compact K\"ahler space with a finite number of singular points, where the metric at each singular point is modelled on an admissible K\"ahler cone. We show that the K\"ahler-Ricci flow with such initial data satisfies a $C/t$ curvature bound, and that the flow near each singular point is modelled on the unique K\"ahler-Ricci expander asymptotic to the corresponding cone. Our motivation is to give a geometric description of the K\"ahler--Ricci flow emerging from singularities arising in the analytic minimal model program.