@article{06d9cf9c9bff4b5f940d50e410854de1,
title = "ON the SHARPNESS of TIAN'S CRITERION for K-STABILITY",
abstract = "Tian's criterion for K-stability states that a Fano variety of dimension n whose alpha invariant is greater than is K-stable. We show that this criterion is sharp by constructing n-dimensional singular Fano varieties with alpha invariants that are not K-polystable for sufficiently large n. We also construct K-unstable Fano varieties with alpha invariants.",
author = "Yuchen Liu and Ziquan Zhuang",
note = "Funding Information: This project was initiated to answer a question of Kento Fujita originated from his paper []. The authors would like to thank Kento Fujita for inspiration and fruitful discussions. The authors would like to thank Harold Blum, Ivan Cheltsov, Christopher Hacon, Chen Jiang, J{\'a}nos Koll{\'a}r, Chi Li, Gang Tian, and Chenyang Xu for helpful discussions and comments. They are also grateful to the anonymous referees for many helpful suggestions and comments. Yuchen Liu would like to thank Sam Payne for his support during the Fall 2018 semester, constant encouragement, and helpful comments. Ziquan Zhuang would also like to thank his advisor J{\'a}nos Koll{\'a}r for constant support and encouragement. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while both authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2019 semester. Publisher Copyright: {\textcopyright} Foundation Nagoya Mathematical Journal, 2020.",
year = "2022",
month = mar,
day = "23",
doi = "10.1017/nmj.2020.28",
language = "English (US)",
volume = "245",
pages = "41--73",
journal = "Nagoya Mathematical Journal",
issn = "0027-7630",
publisher = "Nagoya University",
}