Duke Mathematical Journal

K-stability of cubic threefolds

Yuchen Liu and Chenyang Xu

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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.

Article information

Source
Duke Math. J., Volume 168, Number 11 (2019), 2029-2073.

Dates
Received: 14 June 2017
Revised: 21 December 2018
First available in Project Euclid: 2 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1562033045

Digital Object Identifier
doi:10.1215/00127094-2019-0006

Mathematical Reviews number (MathSciNet)
MR3992032

Zentralblatt MATH identifier
07114913

Subjects
Primary: 14L24: Geometric invariant theory [See also 13A50]
Secondary: 14J70: Hypersurfaces

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

Citation

Liu, Yuchen; Xu, Chenyang. K-stability of cubic threefolds. Duke Math. J. 168 (2019), no. 11, 2029--2073. doi:10.1215/00127094-2019-0006. https://projecteuclid.org/euclid.dmj/1562033045


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