Monge-Ampère equation on compact Hermitian manifolds

Genglong Lin, Yinji Li, Xiangyu Zhou

Published 2023-11-25Version 1

Given a cohomology $(1,1)$-class $\{\beta\}$ of compact Hermitian manifold $(X,\omega)$ such that there exists a bounded potential in $\{\beta\}$, we show that degenerate complex Monge-Amp\`ere equation $(\beta+dd^c \varphi)^n=\mu$ has a unique solution in the full mass class $\mathcal{E}(X,\beta)$, where $\mu$ is any probability measure on $X$ which does not charge pluripolar subset. We also study other Monge-Amp\`ere types equations which correspond to $\lambda>0$ and $\lambda<0$. As a preparation to the $\lambda<0$ case, we give a general answer to an open problem about the Lelong number which was surveyed by Dinew-Guedj-Zeriahi \cite[Problem 36]{DGZ16}. Moreover, we obtain more general results on singular space and of the equations with prescribed singularity if the model potential has small unbounded locus. These results generalize much recent work of \cite{EGZ09}\cite{BBGZ13}\cite{DNL18}\cite{LWZ23} etc.