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

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Vincent Guedj
• Function : Author
Chinh H. Lu
• Function : Author

#### 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.

#### Domains

Mathematics [math]

### Dates and versions

hal-03288391 , version 1 (16-07-2021)

### Identifiers

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

### Cite

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

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