%ACifarelli, Charles%AConlon, Ronan%ADeruelle, Alix%BJournal Name: ArXivorg
%D2022%I
%JJournal Name: ArXivorg
%K
%MOSTI ID: 10332210
%PMedium: X
%TOn finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces
%XWe 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 C×P1 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.
%0Journal Article