Definition 9.10.12FCF157D7001A

Definition 9.10.12.FCF157D7001A A function $\varphi \colon X\rightarrow [-\infty ,\infty )$ is plurisubharmonic if

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   $\varphi $ is not identically $-\infty $ on any irreducible component of $X$, and

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   for any $x\in X$, there is an open neighborhood $V$ of $x$ in $X$, a domain $\Omega \subseteq \mathbb {C}^ N$, a closed immersion $V\hookrightarrow \Omega $ and a plurisubharmonic function $\tilde{\varphi }\in \mathrm{PSH}(\Omega )$ such that $\varphi |_ V=\tilde{\varphi }|_ V$ under this immersion.

The set of plurisubharmonic functions on $X$ is denoted by $\mathrm{PSH}(X)$.

Similarly, if $\theta $ is a smooth closed² real $(1,1)$-form on $X$, then a function $\varphi \colon X\rightarrow [-\infty ,\infty )$ is $\theta $-plurisubharmonic if for any $x\in X$, there is an open neighborhood $V$ of $x$ in $X$, a domain $\Omega \subseteq \mathbb {C}^ N$, a closed immersion $V\hookrightarrow \Omega $ and a smooth function $g$ on $\Omega $ such that $\theta =(\mathrm{dd}^{\mathrm{c}}g)|_ V$ and $g|_ V+\varphi |_ V\in \mathrm{PSH}(V)$.