Singular limits of K\"ahler-Ricci flow on Fano $G$-manifolds

TL;DR

This paper demonstrates that the K"ahler-Ricci flow on certain Fano manifolds without K"ahler-Einstein metrics develops type II singularities, providing the first known examples of such behavior in the literature.

Contribution

It proves that solutions on specific Fano $G$-manifolds are of type II and identifies the first examples of Ricci flow with type II singularities on Fano manifolds.

Findings

Solutions are of type II on certain Fano $G$-manifolds without K"ahler-Einstein metrics.

Constructed explicit examples of Fano compactifications with type II singularities.

First known instances of Ricci flow with type II singularities on Fano manifolds.

Abstract

In this paper, we prove that any solution of K\"ahler-Ricci flow on a Fano compactification $M$ of semisimple complex Lie group, 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.