Horosymmetric limits of Kähler–Ricci flow on Fano $\mathbf{G}$-manifolds

Abstract

We prove that on a Fano $\mathbf{G}$-manifold $(M,J)$, the Gromov–Hausdorff limit of the Kähler–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 Kähler–Ricci soliton. Moreover, $M_{\infty }$ is the limit of a ${\mathbb{C}}^{\ast }$-degeneration of $M$ induced by an element in the Lie algebra of the Cartan torus of $\mathbf{G}$. A similar result can be proved for Kähler–Ricci flows on any Fano horosymmetric manifolds. As an application, we generalize our previous result about the existence of type II singularities of Kähler–Ricci flows on Fano $\mathbf{G}$-manifolds to Fano horosymmetric manifolds.