Lemma 5.24.864CFA1B8522B

Proof. Choose coordinates centered at $p$ and write $\Delta _ R=\{ z\in \mathbb C:|z|<R\} $. On the coordinate polydisk $\Delta _ R^ n$, add a smooth quadratic function to the given function to obtain a plurisubharmonic function $h$ on $\Delta _ R^ n\setminus \{ 0\} $. Write the coordinates as $(z_1,z_2,z')$, with $z'$ absent when $n=2$, and choose $0<r<R$. The compact set

avoids the puncture, so $h\leq M$ on $K$ for some finite $M$. Consider a point of the punctured closed polydisk of radius $r$. If $z_1\ne 0$, the disk in the $z_2$-direction through the point avoids the puncture and has its boundary in $K$, so the maximum principle gives $h\leq M$ at the point. If $z_1=0$ and $(z_2,z')\ne 0$, the disk in the $z_1$-direction through the point avoids the puncture, and its boundary consists of points with $z_1\neq 0$, where the bound has already been proved; the maximum principle again gives $h\leq M$. Subtracting the smooth correction proves the local upper bound.

By lemma 5.24.7, applied with $Z=\{ p\} $, the upper-limit extension belongs to $\mathrm{PSH}(X,\theta )$. It is unique by lemma 5.24.3. If $X$ is compact, the extension is an upper semicontinuous function on a compact space and is therefore bounded above. $\square$