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

TL;DR

This paper proves that the Gromov-Hausdorff limits of K"ahler-Ricci flows on Fano G-manifolds are horosymmetric varieties with singular K"ahler-Ricci solitons, extending previous results to a broader class of manifolds.

Contribution

It establishes the limit behavior of K"ahler-Ricci flows on Fano G-manifolds and horosymmetric manifolds, showing they converge to horosymmetric varieties with solitons, and generalizes previous singularity results.

Findings

Limits are horosymmetric varieties with singular K"ahler-Ricci solitons.

Limits are induced by $ ext{C}^*$-degenerations from the Lie algebra of the Cartan torus.

Results extend to all Fano horosymmetric manifolds.

Abstract

In this paper, we prove that on a Fano $\mathbf G$-manifold $(M,J)$, the Gromov-Hausdorff limit of K\"ahler-Ricci flow with initial metric in $2\pi c_1(M)$ must be a $\mathbb Q$-Fano horosymmetric variety $M_\infty$, which admits a singular K\"ahler-Ricci soliton. Moreover, $M_\infty$ is a limit of $\mathbb C^*$-degeneration of $M$ induced by an element in the Lie algebra of Cartan torus of $\mathbf G$. A similar result can be also proved for K\"ahler-Ricci flows on any Fano horosymmetric manifolds. As an application, we generalize our previous result about the type II singularity of K\"ahler-Ricci flows on Fano $\mathbf G$-manifolds to Fano horosymmetric manifolds.