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-\linebreak 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. 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 Kähler-Ricci flow on compact Kähler surfaces, leading to a classification of the bubbles of such singularities in this dimension.

arXiv comment32 pages; oversight in the proof of Theorem B rectified (uniform lower bound on volume of curves)