A Note On K\"ahler-Ricci Flow on Fano Threefolds

TL;DR

This paper demonstrates that the K"ahler-Ricci flow on certain Fano threefolds develops type II singularities, providing new examples of low-dimensional manifolds with such behavior.

Contribution

It shows that Fano threefolds from a specific family do not admit K"ahler-Ricci solitons and their flow limits are singular varieties, revealing new singularity examples.

Findings

K"ahler-Ricci flow on these threefolds develops type II singularity.

No Fano threefold in the specified family admits a K"ahler-Ricci soliton.

Flow limits are singular $ ext{Q}$-Fano varieties.

Abstract

In this note, we show that the solution of K\"ahler-Ricci flow on every Fano threefold from the family No.2.23 in the Mori-Mukai's list develops type II singularity. In fact, we show that no Fano threefold from the family No.2.23 admits K\"ahler-Ricci soliton and the Gromov-Hausdorff limit of the K\"ahler-Ricci flow must be a singular $\mathbb{Q}$-Fano variety. This gives new examples of Fano manifolds of the lowest dimension on which K\"ahler-Ricci flow develops type II singularity.