Quasi-plurisubharmonic envelopes 2: Bounds on Monge-Amp\`ere volumes

TL;DR

This paper investigates how Monge-Ampère volume measures behave when the reference form is not closed, establishing bounds and a transcendental Grauert-Riemenschneider conjecture related to complex Monge-Ampère equations.

Contribution

It introduces new bounds on Monge-Ampère volumes without the closedness assumption, advancing the understanding of degenerate complex Monge-Ampère equations and related conjectures.

Findings

Established bounds on Monge-Ampère volumes for non-closed reference forms

Proved a transcendental version of the Grauert-Riemenschneider conjecture

Partially answered conjectures of Demailly-Pun and Boucksom-Demailly-Pun-Peternell

Abstract

In \cite{GL21a} we have developed a new approach to $L^{\infty}$-a priori estimates for degenerate complex Monge-Amp\`ere equations, when the reference form is closed. This simplifying assumption was used to ensure the constancy of the volumes of Monge-Amp\`ere 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-P\u{a}un \cite{DP04} and Boucksom-Demailly-P\u{a}un-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-Amp\`ere equations on compact Hermitian varieties.