Extremal omega-plurisubharmonic functions as envelopes of disc functionals

Abstract

For each closed, positive (1,1)-current ωon a complex manifold X and each ω-upper semicontinuous function ϕon X we associate a disc functional and prove that its envelope is equal to the supremum of all ω-plurisubharmonic functions dominated by ϕ. This is done by reducing to the case where ωhas a global potential. Then the result follows from Poletsky's theorem, which is the special case ω=0. Applications of this result include a formula for the relative extremal function of an open set in X and, in some cases, a description of the ω-polynomial hull of a set.

arXiv comment15 pages. Improved notation for the extension of $u+\psi$ over the singular set and refined Definition 2.1 of $\omega$-usc functions.