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15 August 2019 K-stability of cubic threefolds
Yuchen Liu, Chenyang Xu
Duke Math. J. 168(11): 2029-2073 (15 August 2019). DOI: 10.1215/00127094-2019-0006

Abstract

We prove that the K-moduli space of cubic threefolds is identical to their geometric invariant theory (GIT) moduli. More precisely, the K-semistability, K-polystability, and K-stability of cubic threefolds coincide with the corresponding GIT stabilities, which could be explicitly calculated. In particular, this implies that all smooth cubic threefolds admit Kähler–Einstein (KE) metrics and provides a precise list of singular KE ones. To achieve this, our main new contribution is an estimate in dimension 3 of the volumes of Kawamata log terminal singularities introduced by Chi Li. This is obtained via a detailed study of the classification of 3-dimensional canonical and terminal singularities, which was established during the study of the explicit 3-dimensional minimal model program.

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Yuchen Liu. Chenyang Xu. "K-stability of cubic threefolds." Duke Math. J. 168 (11) 2029 - 2073, 15 August 2019. https://doi.org/10.1215/00127094-2019-0006

Information

Received: 14 June 2017; Revised: 21 December 2018; Published: 15 August 2019
First available in Project Euclid: 2 July 2019

zbMATH: 07114913
MathSciNet: MR3992032
Digital Object Identifier: 10.1215/00127094-2019-0006

Subjects:
Primary: 14L24
Secondary: 14J70

Keywords: cubic threefolds , GIT stability , Kähler–Einstein metrics , K-stability , normalized volumes of valuations

Rights: Copyright © 2019 Duke University Press

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Vol.168 • No. 11 • 15 August 2019
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