Algebraic uniqueness of K\"{a}hler-Ricci flow limits and optimal degenerations of Fano varieties

TL;DR

This paper proves the algebraic uniqueness of Kähler-Ricci flow limits on Fano varieties, showing that the Gromov-Hausdorff limit is independent of initial metrics, through new valuation-based functionals.

Contribution

It establishes the uniqueness of special test configurations minimizing the H-functional and confirms a conjecture on the algebraic-uniqueness of Kähler-Ricci flow limits.

Findings

Unique special test configuration minimizing the H-functional.

Confirmation of the algebraic-uniqueness conjecture for flow limits.

Flow limits are independent of initial Kähler metrics.

Abstract

We prove that for any $\mathbb{Q}$-Fano variety $X$, the special $\mathbb{R}$-test configuration that minimizes the $H$-functional is unique and has a K-semistable $\mathbb{Q}$-Fano central fibre $(W, \xi)$. Moreover there is a unique K-polystable degeneration of $(W, \xi)$. As an application, we confirm the conjecture of Chen-Sun-Wang about the algebraic-uniqueness for K\"{a}hler-Ricci flow limits on Fano manifolds, which implies that the Gromov-Hausdorff limit of the flow does not depend on the choice of initial K\"{a}hler metrics. The results are achieved by studying algebraic optimal degeneration problems via new functionals of real valuations, which are analogous to the minimization problem for normalized volumes.