Singular limits of Kähler-Ricci flow on Fano G -manifolds

abstract: Let $M$ be a Fano compactification of a semisimple complex Lie group $G$ and $\omega_0$ a $K\times K$-invariant metric in $2\pi c_1(M)$, where $K$ is a maximal compact subgroup of $G$. Then we prove that the solution of K"ahler-Ricci flow with $\omega_0$ as an initial metric on $M$, is of type II, if $M$ admits no K"ahler-Einstein metrics. As an application, we found two Fano compactifications of $\mathrm{SO}_4(\mathbb{C})$ and one Fano compactification of $\mathrm{Sp}_4(\mathbb{C})$, on which the K"ahler-Ricci flow will develop singularities of type II. To the authors' knowledge, these are the first examples of Ricci flow with singularities of type II on Fano manifolds in the literature.