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

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 Rg whose soliton vector field X has an integral curve along which Rg→0 is biholomorphic to either C¹ 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.