Families of singular K\"ahler-Einstein metrics

Eleonora Di Nezza (1) , Vincent Guedj (2) , Henri Guenancia (3, 2)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 CNRS - Centre National de la Recherche Scientifique

Refining Yau's and Kolodziej's techniques, we establish very precise uniform a priori estimates for degenerate complex Monge-Amp\`ere equations on compact K\"ahler 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 K\"ahler varieties endowed with twisted K\"ahler-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.

Domaines: Mathématiques [math]; Variables complexes [math.CV]; Géométrie différentielle [math.DG]

Eleonora Di Nezza, Vincent Guedj, Henri Guenancia. Families of singular K\"ahler-Einstein metrics. 2020. ⟨hal-02511898v1⟩