Finite Time Singularities of the Kähler Ricci Flow on Compact Kähler Surfaces are of Type I

Abstract

We establish a Type I curvature bound for finite-time volume-collapsing Kähler--Ricci flows on compact Kähler surfaces. No symmetry assumption is imposed, and the initial Kähler class need not be rational. The proof combines local symplectic topology with the classification of gradient Kähler--Ricci shrinking solitons. Bamler's compactness and structure theory provides the shrinking limits used in the argument.