Horosymmetric limits of Kähler-Ricci flow on Fano $G$-manifolds

Gang Tian, Xiaohua Zhu

Published 2022-09-12Version 1

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, we show that $M_\infty$ is a limit of $\mathbb C^*$-degeneration of $(M,J)$ induced by the soliton holomorphic vector field. A similar result can be also proved for K\"ahler-Ricci flows on any Fano horosymmetric manifolds.