Complex Monge–Ampère equations on singular spaces
Annales Polonici Mathematici
MSC: Primary 32U05; Secondary 32W20, 35J66, 35J96
DOI: 10.4064/ap250820-14-2
Published online: 7 May 2026
Abstract
We investigate the complex Monge–Ampère operator on a bounded strongly pseudoconvex domain of a closed, connected, singular, and locally irreducible complex-analytic subvariety. We first examine the classes $\mathcal {E}^p$ for $p \gt 0$ and establish a characterization of their images under the complex Monge–Ampère operator. This result answers a question posed by N. Q. Dieu, T. V. Long. We then turn to the weighted energy classes $\mathcal E_{\chi }(\varOmega )$, consisting of negative plurisubharmonic functions with finite $\chi $-energy, and provide a precise characterization of their images under the complex Monge–Ampère operator, where $\chi $ is a convex increasing function satisfying $\chi (0) = 0$ and $\chi (-\infty ) = -\infty $.
