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Families of singular Kähler-Einstein metrics. (English) Zbl 07714622

Summary: 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.

MSC:

32Q15 Kähler manifolds
14D06 Fibrations, degenerations in algebraic geometry
32Q20 Kähler-Einstein manifolds
32U05 Plurisubharmonic functions and generalizations
32W20 Complex Monge-Ampère operators
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