Finite Time Type I Singularities of the Kähler Ricci Flow

Abstract

We prove the Feldman--Ilmanen--Knopf conjecture for finite time Type I singularities of the Kähler--Ricci flow. More precisely, for any compact Kähler manifold $Y$ and its blow-up $\pi :{\mathrm{Bl}}_{p}Y⟶Y$, if $[{\omega }_0]-T{c}_1(M)={\pi }^{\ast }[{\omega }_Y]$, then any Type I parabolic blow-up limit of the Kähler Ricci flow along the exceptional divisor is the FIK shrinker $\mathrm{Tot}({\mathcal{O}}_{{\mathbb{P}}^{n-1}}(-1))$.

18 pages; updated acknowledgments