Quasi-plurisubharmonic envelopes 2: Bounds on Monge-Amp\`ere volumes
Abstract
In \cite{GL21a} we have developed a new approach to -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.
Keywords
Cite
@article{arxiv.2106.04272,
title = {Quasi-plurisubharmonic envelopes 2: Bounds on Monge-Amp\`ere volumes},
author = {Vincent Guedj and Chinh H. Lu},
journal= {arXiv preprint arXiv:2106.04272},
year = {2023}
}