Pluripolar set

Remarks

Some authors call the sets of the first definition locally pluripolar, and call $E$ globally pluripolar if one function $u$ plurisubharmonic on all of $\C^n$, not identically $-\infty$, has $E \subseteq \set{u = -\infty}$. By Josefson's theorem the two notions agree. Example: $E = \set{z \in \C^2 : z_1 = 0}$ is pluripolar; take $U = \C^2$ and $u(z) = \log\abs{z_1}$, which is plurisubharmonic and equals $-\infty$ exactly on $E$. Non-example: $\R^n \subset \C^n$ is not pluripolar. For $n = 1$: if $u$ is subharmonic near a point of $\R$ and $u = -\infty$ on an interval, then $u \equiv -\infty$ near that point, since intervals are not polar. Applying this on complex lines parallel to the coordinate axes, one coordinate at a time, gives the general case. For $n = 1$, pluripolar sets are exactly polar sets.