General weak limit for Kähler–Ricci flow

Abstract

Consider the Kähler–Ricci flow with finite time singularities over any closed Kähler manifold. We prove the existence of the flow limit in the sense of current toward the time of singularity. This answers affirmatively a problem raised by Tian in [New results and problems on Kähler–Ricci flow, Astérisque 322 (2008) 71–92] on the uniqueness of the weak limit from sequential convergence construction. The notion of minimal singularity introduced by Demailly in the study of positive current comes up naturally. We also provide some discussion on the infinite time singularity case for comparison. The consideration can be applied to more flexible evolution equation of Kähler–Ricci flow type for any cohomology class. The study is related to general conjectures on the singularities of Kähler–Ricci flows.