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ON THE SHARPNESS OF TIAN’S CRITERION FOR K-STABILITY

Part of: Complex manifolds Algebraic groups Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  23 October 2020

YUCHEN LIU  and
ZIQUAN ZHUANG
YUCHEN LIU*
Affiliation:
Department of Mathematics Princeton University Princeton, NJ 08544 USA
ZIQUAN ZHUANG
Affiliation:
Department of Mathematics Massachusetts Institute of Technology Cambridge, Massachusetts 02139 USA ziquan@mit.edu
*
yuchen.liu@yale.edu yuchenl@math.princeton.edu
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Abstract

Tian’s criterion for K-stability states that a Fano variety of dimension n whose alpha invariant is greater than ${n}{/(n+1)}$ is K-stable. We show that this criterion is sharp by constructing n-dimensional singular Fano varieties with alpha invariants ${n}{/(n+1)}$ that are not K-polystable for sufficiently large n. We also construct K-unstable Fano varieties with alpha invariants ${(n-1)}{/n}$.

MSC classification

Primary: 14J45: Fano varieties
Secondary: 14L24: Geometric invariant theory 32Q26: Notions of stability

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Type
Article
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Nagoya Mathematical Journal , Volume 245 , March 2022 , pp. 41 - 73
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Copyright
© Foundation Nagoya Mathematical Journal, 2020

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