PomeDefinition 5.47.2.5458CA6608A5 Let $X$ be a complex manifold. A subset $E\subset X$ is pluripolar, or locally pluripolar, if every point of $X$ has a connected coordinate neighborhood $U$ carrying a function $V\in \mathrm{PSH}(U)$ with $E\cap U\subset \{ V=-\infty \} $. If $X$ is connected, a subset $E\subset X$ is globally pluripolar in $X$ if there is a single $V\in \mathrm{PSH}(X)$ with $E\subset \{ V=-\infty \} $. A property holds quasi-everywhere if it holds outside a pluripolar set.
Definition 5.47.25458CA6608A5
One local pluripolarity definition. The source versions also introduce different related terminology, including global pluripolarity, quasi-everywhere statements, and co-pluripolar sets.
Read the alternative statement and proof
Definition 5.47.2.5458CA6608A5 A subset $E\subseteq X$ is pluripolar if for any $x\in X$, there is an open neighborhood $U$ of $x$ in $X$ and a function $\psi \in \mathrm{PSH}(U)$ such that
\[ \psi |_{E\cap U}\equiv -\infty . \]
A subset $E\subseteq X$ is non-pluripolar if $E$ is not pluripolar.
A subset $F\subseteq X$ is co-pluripolar if $X\setminus F$ is pluripolar.
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