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