Optimal destabilization of K-unstable Fano varieties via stability thresholds

We show that for a K-unstable Fano variety, any divisorial valuation computing its stability threshold induces a nontrivial 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, Szekelyhidi 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.

Automated summary

This paper proves that for a K-unstable Fano variety, any divisorial valuation computing its stability threshold induces a nontrivial special test configuration preserving the stability threshold. When such a divisorial valuation exists, the Fano variety degenerates to a uniquely determined twisted K-polystable Fano variety. The stability threshold can also be approximated by divisorial valuations induced by special test configurations.