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 of the volumes of Kawamata log terminal singularities introduced by Chi Li. This is obtained via a detailed study of the classification of -dimensional canonical and terminal singularities, which was established during the study of the explicit -dimensional minimal model program.
"K-stability of cubic threefolds." Duke Math. J. 168 (11) 2029 - 2073, 15 August 2019. https://doi.org/10.1215/00127094-2019-0006