K-stability of cubic fourfolds

Abstract

We prove that the K-moduli space of cubic fourfolds is identical to their GIT moduli space. More precisely, the K-(semi/poly)stability of cubic fourfolds coincide to the corresponding GIT stabilities, which was studied in detail by Laza. In particular, this implies that all smooth cubic fourfolds admit Kähler-Einstein metrics. Key ingredients are local volume estimates in dimension three due to Liu-Xu, and Ambro-Kawamata's non-vanishing theorem for Fano fourfolds.

21 pages. Comments welcome. v2: exposition improved, to appear in J. Reine Angew. Math. (Crelle's Journal)

References in corpus (6)

• CM Stability and the Generalized Futaki Invariant I
• Optimal destabilizing centers and equivariant K-stability
• K-stability of log Fano hyperplane arrangements
• Uniqueness of the minimizer of the normalized volume function
• Equivariant K-stability under finite group action
• K-moduli of curves on a quadric surface and K3 surfaces

Cited by in corpus (5)

• K-moduli of curves on a quadric surface and K3 surfaces
• K-stability and birational models of moduli of quartic K3 surfaces
• Cohomology of moduli space of cubic fourfolds I
• On the continuity of Weil-Petersson volumes of the moduli space weighted points on the projective line
• Log K-stability of GIT-stable divisors on Fano varieties