Quasi-plurisubharmonic envelopes 2: Bounds on Monge-Ampère volumes

Abstract

In \cite{GL21a} we have developed a new approach to $L^{\infty }$-a priori estimates for degenerate complex Monge-Ampère equations, when the reference form is closed. This simplifying assumption was used to ensure the constancy of the volumes of Monge-Ampère measures. We study here the way these volumes stay away from zero and infinity when the reference form is no longer closed. We establish a transcendental version of the Grauert-Riemenschneider conjecture, partially answering conjectures of Demailly-Păun \cite{DP04} and Boucksom-Demailly-Păun-Peternell \cite{BDPP13}. Our approach relies on a fine use of quasi-plurisubharmonic envelopes. The results obtained here will be used in \cite{GL21b} for solving degenerate complex Monge-Ampère equations on compact Hermitian varieties.

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• Intrinsic capacities on compact Kähler manifolds
• Blow-ups and resolutions of strong Kähler with torsion metrics
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• On the invariance of the total Monge-Ampere volume of Hermitian metrics
• Fully non-linear degenerate elliptic equations in complex geometry

Cited by in corpus (1)

• Plurisupported Currents on Compact Kähler Manifolds