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I came across a problem like this. Suppose that $\Omega$ is an open subset of $\mathbb{C}^{n}$ and $V$ is a complex submanifold of $\Omega$ of codimension 1. Now given a plurisubharmonic function $\varphi$ on $\Omega-V$ which is bounded in the sense that given any compact set $K\subset\Omega$, $\varphi$ is bounded above in $K-V$, can we claim that we can extend $\varphi$ to be a plurisubharmonic function on $\Omega$? And are there any reference about extending a plurisubharmonic function in more general case?

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$V$ is pluripolar. Let $v$ be a plurisubharmonic function which is $-\infty$ on $V$. Then $\phi+\epsilon v$ is plurisubharmonic for $\epsilon>0$ (the definition of plurisubharmonic function is easily verified for it). Therefore when $\epsilon\to 0$ the limit must be plurisubharmonic.

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