Can a complete pluripolar set be a single point?

Question

Let $f:\mathbb{C}^n\rightarrow\mathbb{R}\cup\{-\infty\}$ be a plurisubharmonic function which is not identically $-\infty$.The set $\mathcal{P}:=\{z\in\mathbb{C}^n:f(z)=-\infty\}$ is called a complete pluripolar set.

My question is:is it posibble that $\mathcal{P}$ is a single point?Why?

Any answers are appreciated.Thanks in advance!

Answer 1

Yes, every finite set is complete pluripolar. If $E = \{ w_1, \ldots, w_n \}$, take for example $$ u(z) = \sum_{j=1}^n \log \| z-w_j \|. $$