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Optimal destabilization
of K–unstable Fano varieties via stability thresholds
Harold Blum, Yuchen Liu and Chuyu Zhou |
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Geometry & Topology 26 (2022) 2507–2564
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DOI: 10.2140/gt.2022.26.2507
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Abstract |
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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. |
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Keywords
K-stability, Fano varieties, optimal destabilization
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Mathematical Subject Classification 2010
Primary: 14J10, 14J45, 32Q20
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References |
Publication
Received: 21 March 2020
Revised: 14 May 2021
Accepted: 7 July 2021
Published: 13 December 2022
Proposed: Gang Tian
Seconded: Dan Abramovich, Simon Donaldson
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Authors |
| Harold Blum | |
| Department of Mathematics Stony Brook University Stony Brook, NY United States |
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| Yuchen Liu | |
| Department of Mathematics Northwestern University Evanston, IL United States |
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| Chuyu Zhou | |
| Institute of Mathematics Ecole Polytechnique Federale de Lausanne (EPFL) Lausanne Switzerland |
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