PomeDefinition 5.8.1.6070D2D83EC6 Let $\Omega \subset \mathbb {C}^ n$ be a domain. If $n\ge 1$, a function $u\colon \Omega \to [-\infty ,\infty )$ is plurisubharmonic if $u\not\equiv -\infty $, $u$ is upper semicontinuous, and for every affine complex line $L$ the restriction of $u$ to each connected component of $L\cap \Omega $ is subharmonic or identically $-\infty $. The set of plurisubharmonic functions on $\Omega $ is denoted by $\mathrm{PSH}(\Omega )$. In dimension zero a domain is a single point, and the plurisubharmonic functions on it are the real-valued functions.
Definition 5.8.16070D2D83EC6
The same line-restriction definition, including dimension zero. Source explanations and numbered clauses remain available.
Read the alternative statement and proof
Definition 5.8.1.6070D2D83EC6 When $n\geq 1$, a plurisubharmonic function on $\Omega $ is a function $\varphi \colon \Omega \rightarrow [-\infty ,\infty )$ satisfying the following three conditions:
1E4DADA74A2A$\varphi \not\equiv -\infty $;
62EDD3578A72$\varphi $ is upper semicontinuous;
B250529A16B9for any complex line $L\subseteq \mathbb {C}^ n$ and any connected component $U$ of $L\cap \Omega $, the restriction $\varphi |_ U$ is either subharmonic or constantly $-\infty $.1
When $n=0$, the only domain $\Omega $ is the singleton. In this case, a plurisubharmonic function on $\Omega $ is a real-valued function on $\Omega $.
The set of plurisubharmonic functions on $\Omega $ is denoted by $\mathrm{PSH}(\Omega )$.
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