Tong-Xin Xu

Finite Time Type I Singularities of the K\"ahler Ricci Flow

We prove the Feldman--Ilmanen--Knopf conjecture for finite time Type I singularities of the K\"ahler--Ricci flow. More precisely, for any compact K\"ahler manifold $Y$ and its blow-up $\pi:\operatorname{Bl}_pY\longrightarrow Y$, if $[\omega_0]-Tc_1(M)=\pi^*[\omega_Y]$, then any Type I parabolic blow-up limit of the K\"ahler Ricci flow along the exceptional divisor is the FIK shrinker $\operatorname{Tot}(\mathcal{O}_{\mathbb{P}^{n-1}}(-1))$.