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On the Range of a class of Complex Monge-Ampère operators on compact Hermitian manifolds (2404.03246v1)
Published 4 Apr 2024 in math.CV
Abstract: Let $(X,\omega)$ be a compact Hermitian manifold of complex dimension $n$. Let $\beta$ be a smooth real closed $(1,1)$ form such that there exists a function $\rho \in \mbox{PSH}(X,\beta)\cap L{\infty}(X)$. We study the range of the complex non-pluripolar Monge-Amp`ere operator $\langle(\beta+ddc\cdot)n\rangle$ on weighted Monge-Amp`ere energy classes on $X$. In particular, when $\rho$ is assumed to be continuous, we give a complete characterization of the range of the complex Monge-Amp`ere operator on the class $\mathcal E(X,\beta)$, which is the class of all $\varphi \in \mbox{PSH}(X,\beta)$ with full Monge-Amp`ere mass, i.e. $\int_X\langle (\beta+ddc\varphi)n\rangle=\int_X\betan$.