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

Vincent Guedj (1) , Chinh H. Lu (2, 3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 LMO - Laboratoire de Mathématiques d'Orsay
3 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

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.

Mots clés: June 27; 2021. 2010 Mathematics Subject Classification. 32W20; 32U05; 32Q15; 35A23 Monge-Ampère equation; a priori estimates; June 27

Domaines: Mathématiques [math]

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