# 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-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.
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https://hal-universite-paris-saclay.archives-ouvertes.fr/hal-03288391
Contributor : Hoang-Chinh Lu Connect in order to contact the contributor
Submitted on : Friday, July 16, 2021 - 11:55:34 AM
Last modification on : Tuesday, January 4, 2022 - 5:58:46 AM
Long-term archiving on: : Sunday, October 17, 2021 - 6:35:36 PM

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Quasipshenvelopes2.pdf
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### Identifiers

• HAL Id : hal-03288391, version 1
• ARXIV : 2106.04272

### Citation

Vincent Guedj, Chinh H. Lu. Quasi-plurisubharmonic envelopes 2: Bounds on Monge-Ampère volumes. 2021. ⟨hal-03288391⟩

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