@article{3eab7ad64eae4e6f89026baa276651ce,
title = "Uniqueness of K-polystable degenerations of Fanovarieties",
abstract = "We prove that K-polystable degenerations of ℚ -Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable ℚ-Fano varieties is separated. Together with recently proven boundedness and openness statements, the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly K-stable ℚ-Fano varieties of fixed dimension and volume. The result also implies that the automorphism group of a K-stable ℚ-Fano variety is finite.",
keywords = "Degenerations, Fano varieties, K-stability, Moduli",
author = "Harold Blum and Chenyang Xu",
note = "Funding Information: HB is partially supported by NSF grant DMS-1803102. CX is partially supported by the National Science Fund for Distinguished Young Scholars (NSFC 11425101) “Algebraic Geometry.” Funding Information: We are grateful to Roman Bezrukavnikov, Giulio Codogni, Tommaso de Fernex, Christopher Hacon, Mattias Jonsson, J{\'a}nos Koll{\'a}r, Chi Li, Yuchen Liu, Mircea Mustatagrave;, Yuji Odaka, Xiaowei Wang, and un Yu for helpful conversations and comments on previous drafts of this paper. A large part of the work on this paper was completed when CX was visiting Institut Henri Poincar{\'e} as part of the Poincar{\'e} Chair program. He thanks the institute for the wonderful environment and Claire Voisin for her hospitality. Finally, we are indebted to the anonymous referees whose comments improved this paper considerably. HB is partially supported by NSF grant DMS-1803102. CX is partially supported by the National Science Fund for Distinguished Young Scholars (NSFC 11425101) \Algebraic Geometry.{"} Publisher Copyright: {\textcopyright} 2019 Department of Mathematics, Princeton University.",
year = "2019",
doi = "10.4007/annals.2019.190.2.4",
language = "English (US)",
volume = "190",
pages = "609--656",
journal = "Annals of Mathematics",
issn = "0003-486X",
publisher = "Princeton University Press",
number = "2",
}