Kähler-Ricci flow on $\mathbf{G}$-spherical Fano manifolds

Abstract

We prove that the Gromov-Hausdorff limit of Kähler-Ricci flow on a $\mathbf{G}$-spherical Fano manifold $X$ is a $\mathbf{G}$-spherical $\mathbb{Q}$-Fano variety $X_{\infty }$, which admits a (singular) Kähler-Ricci soliton. Moreover, the $\mathbf{G}$-spherical variety structure of $X_{\infty }$ can be constructed as a center of torus ${\mathbb{C}}^{\ast }$-degeneration of $X$ induced by an element in the Lie algebra of Cartan torus of $\mathbf{G}$.