K-stability of cubic fourfolds

Yuchen Liu

2022Journal für die reine und angewandte Mathematik (Crelles Journal)13 citationsDOI

Abstract

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 and Xu, and Ambro–Kawamata’s non-vanishing theorem for Fano fourfolds.

Topics & Concepts

Moduli spaceFano planeDimension (graph theory)Stability (learning theory)Pure mathematicsMathematicsSpace (punctuation)Mathematical physicsComputer scienceOperating systemMachine learningGeometry and complex manifoldsGeometric Analysis and Curvature FlowsAlgebraic Geometry and Number Theory