Quasi-plurisubharmonic envelopes 1: uniform estimates on Kähler manifolds

Abstract

We develop a new approach to a priori $L^{\infty }$-estimates for degenerate complex Monge–Ampère equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. Our method allows one to obtain new and efficient proofs of several fundamental results in Kähler geometry. In a sequel we shall explain how this approach also applies to the hermitian setting, producing new relative a priori bounds, as well as existence results.