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Optimal destabilization of K–unstable Fano varieties via stability thresholds

Harold Blum, Yuchen Liu and Chuyu Zhou

Geometry & Topology 26 (2022) 2507–2564
DOI: 10.2140/gt.2022.26.2507

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, 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.

K-stability, Fano varieties, optimal destabilization
Mathematical Subject Classification 2010
Primary: 14J10, 14J45, 32Q20
Received: 21 March 2020
Revised: 14 May 2021
Accepted: 7 July 2021
Published: 13 December 2022
Proposed: Gang Tian
Seconded: Dan Abramovich, Simon Donaldson
Harold Blum
Department of Mathematics
Stony Brook University
Stony Brook, NY
United States
Yuchen Liu
Department of Mathematics
Northwestern University
Evanston, IL
United States
Chuyu Zhou
Institute of Mathematics
Ecole Polytechnique Federale de Lausanne (EPFL)