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

Abstract: We develop a new approach to $L^{\infty}$-a priori estimates for degenerate complex Monge-Amp`ere equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. In a prequel \cite{GL21a} we have shown how this method allows one to obtain new and efficient proofs of several fundamental results in K\"ahler geometry. In \cite{GL21b} we have studied the behavior of Monge-Amp`ere volumes on hermitian manifolds. We extend here the techniques of \cite{GL21a} to the hermitian setting and use the bounds established in \cite{GL21b}, producing new relative a priori estimates, as well as several existence results for degenerate complex Monge-Amp`ere equations on compact hermitian manifolds.