Generalized optimal degenerations of Fano varieties

We prove a generalization of the algebraic version of Tian conjecture. Precisely, for any smooth strictly increasing function $g:\mathbb{R}\to {\mathbb{R}}_{>0}$ with $\mathrm{l}\mathrm{o}\mathrm{g}\circ g$ convex, we define the ${\mathbf{H}}^g$-invariant on a Fano variety $X$ generalizing the $\mathbf{H}$-invariant introduced by Tian-Zhang-Zhang-Zhu, and show that ${\mathbf{H}}^g$ admits a unique minimizer. Such a minimizer will induce the $g$-optimal degeneration of the Fano variety $X$, whose limit space admits a $g^{\mathrm{\prime }}$-soliton. We present an example of Fano threefold which has the same $g$-optimal degenerations for any $g$.