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.