Quasi-plurisubharmonic envelopes 3: Solving Monge–Ampère equations on hermitian manifolds

Abstract

Abstract We develop a new approach to L ∞ L^{\infty} -a priori estimates for degenerate complex Monge–Ampère equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. In a prequel [Quasi-plurisubharmonic envelopes 1: Uniform estimates on Kähler manifolds, preprint (2021), https://arxiv.org/abs/2106.04273], we have shown how this method allows one to obtain new and efficient proofs of several fundamental results in Kähler geometry. In [Quasi-plurisubharmonic envelopes 2: Bounds on Monge–Ampère volumes, Algebr. Geom. 9 (2022), 6, 688–713], we have studied the behavior of Monge–Ampère volumes on hermitian manifolds. We extend here the techniques of the former to the hermitian setting and use the bounds established in the latter, producing new relative a priori estimates, as well as several existence results for degenerate complex Monge–Ampère equations on compact hermitian manifolds.