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 R_gwhose soliton vector field X has an integral curve along which R_g→ 0 is biho-lomorphic to either C × P¹or 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.