Generalized optimal degenerations of Fano varieties

TL;DR

This paper generalizes Tian's conjecture by introducing a new invariant for Fano varieties, proving it has a unique minimizer, and exploring the resulting optimal degenerations and their properties.

Contribution

It introduces the $ extbf{H}^g$-invariant for Fano varieties, generalizes the algebraic Tian conjecture, and establishes the existence and uniqueness of $g$-optimal degenerations.

Findings

The $ extbf{H}^g$-invariant admits a unique minimizer.

The minimizer induces a $g$-optimal degeneration with a $g'$-soliton limit.

Existence of Fano threefolds with identical $g$-optimal degenerations for all $g$.

Abstract

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 ${\rm log}\circ g$ 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 $g'$-soliton. We present an example of Fano threefold which has the same $g$-optimal degenerations for any $g$.