On finite time Type I singularities of the Kähler–Ricci flow on compact Kähler surfaces

Abstract

We show that the underlying complex manifold of a complete non-compact two-dimen­sional shrinking gradient Kähler–Ricci soliton $(M,g,X)$ with soliton metric $g$ with bounded scalar curvature $R_g$ whose soliton vector field $X$ has an integral curve along which $R_g\nrightarrow 0$ is biholomorphic to either $\mathbb{C}\times {\mathbb{P}}^1$ or to the blowup of this manifold at one point, and that the soliton metric $g$ is toric. We also identify the corresponding soliton vector field $X$ in each case. Given these possibilities, we then prove a strong form of the Feldman–Ilmanen–Knopf conjecture for finite time Type I singularities of the Kähler–Ricci flow on compact Kähler surfaces, leading to a classification of the bubbles of such singularities in this dimension.