Extremal omega-plurisubharmonic functions as envelopes of disc functionals - Generalization and applications to the local theory

TL;DR

This paper extends the Poletsky disc envelope formula to a broader class of omega-plurisubharmonic functions on complex manifolds, involving differences of positive currents and functions.

Contribution

It generalizes the classical envelope formula to cases where the current and functions are differences of positive currents and upper semicontinuous functions.

Findings

Extended envelope formula to complex manifolds with difference of positive currents.

Provided applications to local theory of omega-plurisubharmonic functions.

Broadened the scope of disc envelope representations in complex analysis.

Abstract

We generalize the Poletsky disc envelope formula for the function $\sup \{u \in \PSH(X,\omega) ; u\leq \phi\}$ on any complex manifold $X$ to the case where the real (1,1)-current $\omega=\omega_1-\omega_2$ is the difference of two positive closed (1,1)-currents and $\phi$ is the difference of an $\omega_1$-upper semicontinuous function and a plurisubharmonic function.