PomeCorollary 9.3.18.6DDD0384D260 Let $A$ be an analytic subset of codimension at least two of a complex manifold $M$. Then every plurisubharmonic function $u$ on $M\setminus A$ is locally bounded above near $A$ and extends uniquely to a plurisubharmonic function on $M$, whose value at $a\in A$ is $\varlimsup _{M\setminus A\ni y\to a}u(y)$.
Corollary 9.3.186DDD0384D260
Proof. The assertion is local, and extensions defined on overlapping charts agree, since their values on $A$ are given by the same formula. Near a point where $M$ has dimension at most one, the codimension condition forces $A=\varnothing $. Otherwise, let $\Omega $ be a connected coordinate chart of dimension $n\geq 2$. Then $\dim A\leq n-2$, and corollary 9.3.13(3) shows that $A\cap \Omega $ is locally $\mathcal H^{2n-3}$-null, since $2n-3>2(n-2)$. By theorem 5.11.3, $u$ extends uniquely to a plurisubharmonic function on $\Omega $ with the stated values on $A$; being upper semicontinuous, the extension is locally bounded above. $\square$
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