In this paper, we prove an existence result for Kähler-Einstein metrics on $\mathbb{Q}$-Fano compactifications of Lie groups by the variational method, provided their moment polytopes satisfy a fine condition. As an application, we prove that there is no $\mathbb{Q}$-Fano ${\mathrm{S}\mathrm{O}}_4(\mathbb{C})$-compactification which admits a Kähler-Einstein metric with the same volume as that of a smooth K-unstable Fano ${\mathrm{S}\mathrm{O}}_4(\mathbb{C})$-compactification.

We prove that the Gromov–Hausdorff limit of Kähler–Ricci flow on a G-spherical Fano manifold X is a G-spherical $\mathbb{Q}$-Fano variety X_∞, which admits a (singular) Kähler–Ricci soliton. Moreover, the G-spherical variety structure of X_∞ 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 G.

In this paper, we prove the existence of solutions to the Fu–Yau equation on compact Kähler manifolds. As an application, we give a class of non-trivial solutions of the modified Strominger system.