On finite time Type I singularities of the K\"ahler-Ricci flow on compact K\"ahler surfaces

TL;DR

This paper classifies finite time Type I singularities of the K"ahler-Ricci flow on compact K"ahler surfaces, identifying possible singularity models and proving a strong conjecture in this setting.

Contribution

It provides a classification of singularity models for the K"ahler-Ricci flow on surfaces, proving a strong form of the Feldman-Ilmanen-Knopf conjecture.

Findings

Singularity models are biholomorphic to   or its blowup.

The soliton on the blowup is toric and unique.

Identifies the soliton vector field associated with these models.

Abstract

We show that the underlying complex manifold of a complete non-compact two-\linebreak dimensional shrinking gradient K\"ahler-Ricci soliton $(M,\,g,\,X)$ with soliton metric $g$ with bounded scalar curvature $\operatorname{R}_{g}$ whose soliton vector field $X$ has an integral curve along which $\operatorname{R}_{g}\not\to0$ is biholomorphic to either $\mathbb{C}\times\mathbb{P}^{1}$ or to the blowup of this manifold at one point. Assuming the existence of such a soliton on this latter manifold, we show that it is toric and unique. We also identify the corresponding soliton vector field. Given these possibilities, we then prove a strong form of the Feldman-Ilmanen-Knopf conjecture for finite time Type I singularities of the K\"ahler-Ricci flow on compact K\"ahler surfaces, leading to a classification of the bubbles of such singularities in this dimension.