Reading Demailly's Algebraic and Complex geometry in particular chapter 3 about positive currents, tha author defines the unbounded locus $L(u)$ of a plurisubharmonic function $u$ to be the set of points $x\in X$ such that $u$ is unbounded in every neighborhood of $x$. Then he shows that $L(u)$ contains the closure of $P(u)=u^{1}(-\infty)$, but in general this inclusion is strict: in fact, $u(z)=\sum k^{-2}\log(|z-1/k|+e^{-k^{3}})$ is everywhere finite in $\mathbb{C}$ but $L(u)=(0)$. what i don't understand is why is everywhere finite in $\mathbb{C}$, since if $z=1/k$ then $u(z)=-\infty$
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1$\begingroup$ No, when $k$ is fixed, $u(1/k)$ is not $-\infty$: it is the sum of $-k$ and some convergent series. The point is that near $0$ you can find arbitrary negative values of $u$ though $u$ has no pole. $\endgroup$– HenriCommented Dec 11, 2011 at 20:51
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$\begingroup$ thank you...so if $u$ has logaritmic singularities then $u^{-1}(-\infty)=L(u)$? is it correct? $\endgroup$– alikeCommented Dec 12, 2011 at 16:57
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$\begingroup$ Yes, it is correct! $\endgroup$– HenriCommented Dec 12, 2011 at 19:46
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