Families of singular Kähler–Einstein metrics

Abstract

Refining Yau’s and Kołodziej’s techniques, we establish very precise uniform a priori estimates for degenerate complex Monge–Ampère equations on compact Kähler manifolds, that allow us to control the blow up of the solutions as the cohomology class and the complex structure both vary. We apply these estimates to the study of various families of possibly singular Kähler varieties endowed with twisted Kähler–Einstein metrics, by analyzing the behavior of canonical densities, establishing uniform integrability properties, and developing the first steps of a pluripotential theory in families. This provides interesting information on the moduli space of stable varieties, extending works by Berman–Guenancia and Song, as well as on the behavior of singular Ricci-flat metrics on (log) Calabi–Yau varieties, generalizing works by Rong–Ruan–Zhang, Gross–Tosatti–Zhang, Collins–Tosatti and Tosatti–Weinkove–Yang.