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arXiv · 2404.03246

On the Range of a class of Complex Monge-Ampère operators on compact Hermitian manifolds

Yinji Li·Zhiwei Wang·Xiangyu Zhou
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Abstract

Let $(X,ω)$ be a compact Hermitian manifold of complex dimension $n$. Let $β$ be a smooth real closed $(1,1)$ form such that there exists a function $ρ\in \mbox{PSH}(X,β)\cap L^{\infty}(X)$. We study the range of the complex non-pluripolar Monge-Ampère operator $\langle(β+dd^c\cdot)^n\rangle$ on weighted Monge-Ampère energy classes on $X$. In particular, when $ρ$ is assumed to be continuous, we give a complete characterization of the range of the complex Monge-Ampère operator on the class $\mathcal E(X,β)$, which is the class of all $φ\in \mbox{PSH}(X,β)$ with full Monge-Ampère mass, i.e. $\int_X\langle (β+dd^cφ)^n\rangle=\int_Xβ^n$.

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Yinji Li, Zhiwei Wang, Xiangyu Zhou. 2024-04-04. On the Range of a class of Complex Monge-Ampère operators on compact Hermitian manifolds. https://arxiv.org/abs/2404.03246

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