Optimal destabilization of K-unstable Fano varieties via stability thresholds

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, Székelyhidi, 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.

Final version, to appear in Geom. Topol

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Cited by in corpus (9)

• K-stability of Fano varieties: an algebro-geometric approach
• On properness of K-moduli spaces and optimal degenerations of Fano varieties
• Continuity of delta invariants and twisted Kähler--Einstein metrics
• Equivariant K-stability under finite group action
• Toward finite generation of higher rational rank valuations
• On wall crossing for K-stability
• On K-stability of Calabi-Yau fibrations
• Log K-stability of GIT-stable divisors on Fano varieties
• Approximating delta invariants in the sense of complements of plt type