# Optimal destabilization of K-unstable Fano varieties via stability thresholds

## TL;DR

This paper investigates the destabilization process of K-unstable Fano varieties, showing how valuations and test configurations relate to their stability thresholds and degenerations, with implications for Ricci bounds.

## Contribution

It introduces a method linking divisorial valuations to special test configurations, leading to a unique degeneration to twisted K-polystable Fano varieties.

## Key findings

- Divisorial valuations induce non-trivial test configurations.
- Fano varieties degenerate to twisted K-polystable forms.
- Ricci lower bounds form a finite set of rational numbers.

## Abstract

We show that for a K-unstable Fano variety, any divisorial valuation computing its stability threshold induces a non-trivial special test configuration preserving the stability threshold. When such a divisorial valuation exists, we show that the Fano variety degenerates to a uniquely determined twisted K-polystable Fano variety. We also show that the stability threshold can be approximated by divisorial valuations induced by special test configurations. As an application of the above results and the analytic work of Datar, Sz\'ekelyhidi, and Ross, we deduce that greatest Ricci lower bounds of Fano manifolds of fixed dimension form a finite set of rational numbers. As a key step in the proofs, we adapt the process of Li and Xu producing special test configurations to twisted K-stability in the sense of Dervan.