Definition 5.17.13DC2C56EBB31

Definition 5.17.1.3DC2C56EBB31 Let $\theta $ be a closed real smooth $(1,1)$-form on $X$.

A $\theta $-plurisubharmonic function or $\theta $-psh function for short on $X$ is a function $\varphi \colon X\rightarrow [-\infty ,\infty )$ such that for each $x\in X$ and each open neighborhood $U$ of $x$ in $X$ satisfying the condition that $\theta =\mathrm{dd}^{\mathrm{c}}g$ for some smooth function $g$ on $U$, we have $g+\varphi |_ U\in \mathrm{PSH}(U)$. The set of $\theta $-psh functions on $X$ is denoted by $\mathrm{PSH}(X,\theta )$.

A quasi-plurisubharmonic function on $X$ is a function $\varphi \colon X\rightarrow [-\infty ,\infty )$ such that there exists a smooth closed real $(1,1)$-form $\theta '$ on $X$ such that $\varphi \in \mathrm{PSH}(X,\theta ')$. The set of quasi-plurisubharmonic functions on $X$ is denoted by $\mathrm{QPSH}(X)$.