Local noncollapsing for complex Monge-Ampère equations (Q2100861)

This paper proves a local non-collapsing result on the relative volume, under a local Ricci lower bound on a geodesic ball, and some very weak integrability condition on the volume density. The Kähler class is allowed to degenerate. As applications, the authors reprove a uniform diameter bound. Other applications include Kähler-Ricci flows, and a local gradient bound on the Kähler potential. The main techniques come from the recent PDE proof of Kolodziej's \(C^0\) potential estimate, inspired by Chen-Cheng's seminal work. Other ingredients come from comparison theorems in Riemannian geometry.