Monge-Ampère operators on compact Kähler manifolds
Vincent Guedj, Ahmed Zeriahi
TL;DR
The paper studies the complex Monge-Ampère operator on compact Kähler manifolds and seeks to describe the range of ω_φ^2 on ω-psh functions with L^2 gradient and finite energy, generalizing local pluripotential theory to the compact setting. It introduces energy classes E(X,ω) = PSH(X,ω) ∩ W^{1,2}(X) and the higher-energy classes E^p(X,ω), analyzes convergence of associated currents, and establishes stability and capacity properties relevant to the Monge-Ampère equation. The main result characterizes when a probability measure μ lies in the range of the operator: μ = ω_ψ^2 for a unique ψ ∈ E^p(X,ω) with sup_X ψ = -1 if and only if E^p(X,ω) ⊂ L^p(μ); existence is shown via Aubin–Yau approximations. The framework extends to higher dimensions and has applications to complex dynamics and to constructing Kähler-Einstein metrics on singular manifolds, providing a robust approach to degenerate Monge-Ampère equations on compact Kähler spaces.
Abstract
We study the complex Monge-Amp\` ere operator on compact K\"ahler manifolds. We give a complete description of its range on the set of $\omega-$plurisubharmonic functions with $L^2$ gradient and finite self energy, generalizing to this compact setting results of U.Cegrell from the local pluripoltential theory. We give some applications to complex dynamics and to the existence of K\"ahler-Einstein metrics on singular manifolds.