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 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.