## Duke Mathematical Journal

### K-stability of cubic threefolds

#### 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
Revised: 21 December 2018
First available in Project Euclid: 2 July 2019

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
Secondary: 14J70: Hypersurfaces

#### 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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