Theorem 5.11.376D63A606619

Proof. Let $u$ be a plurisubharmonic function on $\Omega \setminus S$. For every closed ball $\overline B\subset \Omega $, the compact set $S\cap \overline B$ is covered by finitely many neighborhoods $W$ as in the hypothesis, and is therefore $\mathcal H^{2n-3}$-null.

Step 1: $u$ is locally bounded above near $S$. Fix $p\in S$ and $r>0$ with $\overline B=\overline{B(p,r)}\subset \Omega $, and write $B=B(p,r)$. By lemma 5.11.2(2) applied with $x=p$, almost every complex line through $p$ meets $S\cap \overline B$ only at $p$; fix such a line $L=\{ p+te:t\in \mathbb {C}\} $, with $|e|=1$. The circle $\Gamma =L\cap \partial B$ is a compact subset of $\Omega \setminus S$, so the upper semicontinuous function $u$ is bounded above on $\Gamma $ by a finite constant $C$, and the open set $O=\{ y\in \Omega \setminus S:u(y)<C+1\} $ contains $\Gamma $.

Suppose that $u$ is not bounded above near $p$. Then there are points $x_ j\in B\setminus S$ with $x_ j\to p$ and $u(x_ j)\to +\infty $. For each $j$, lemma 5.11.2(2) applied with $x=x_ j$ shows that the directions of the lines through $x_ j$ meeting $S\cap \overline B$ form a $\sigma $-null set. Every nonempty open ball in $\mathbb P^{n-1}(\mathbb C)$ has positive $\sigma $-measure by lemma 2.9.19, so the complement of this null set is dense. We can therefore choose directions $\xi _ j$ outside this null set with $d(\xi _ j,[e])<1/j$. Since $d$ is the minimum of the spherical distance over the unit representatives, $\xi _ j$ has a unit representative $e_ j$ whose spherical distance to $e$ is less than $1/j$, and then $|e_ j-e|<1/j$, because a chord is shorter than the arc. Thus $e_ j\to e$, and the lines $L_ j=\{ x_ j+te_ j:t\in \mathbb {C}\} $ do not meet $S\cap \overline B$.

We claim that $L_ j\cap \partial B\subset O$ for all large $j$. Let $c_ j$ be the point of $L_ j$ closest to $p$, that is, the orthogonal projection of $p$ to $L_ j$; then $|c_ j-p|\leq |x_ j-p|\to 0$. Since $L_ j$ is a complex line, $c_ j=x_ j+\langle p-x_ j,e_ j\rangle e_ j$ and $\langle c_ j-p,e_ j\rangle =0$ for the Hermitian inner product, so $\operatorname{Re}\langle c_ j-p,se_ j\rangle =0$ and $|c_ j+se_ j-p|^2=|c_ j-p|^2+|s|^2$ for $s\in \mathbb {C}$. Hence $L_ j\cap \partial B$ is the circle $\{ c_ j+\rho _ j\mathrm{e}^{\mathrm{i}\theta }e_ j\} $ with $\rho _ j=(r^2-|c_ j-p|^2)^{1/2}\to r$. Since $|c_ j+\rho _ j\mathrm{e}^{\mathrm{i}\theta }e_ j-p-r\mathrm{e}^{\mathrm{i}\theta }e|\leq |c_ j-p|+|\rho _ j-r|+r|e_ j-e|$, these circles converge uniformly in $\theta $ to $\Gamma =\{ p+r\mathrm{e}^{\mathrm{i}\theta }e\} $. The distance from the compact set $\Gamma $ to the closed complement of the open set $O$ is positive, so the circles lie in $O$ for large $j$.

For such $j$, the closed disc $\Delta _ j=L_ j\cap \overline B$ lies in $\Omega \setminus S$, and the restriction of $u$ to the connected component of $L_ j\cap (\Omega \setminus S)$ containing $\Delta _ j$ is subharmonic or identically $-\infty $. Its values on the boundary circle of $\Delta _ j$ are less than $C+1$, so the maximum principle, that is, theorem 3.4.5 with the constant harmonic function $C+1$, gives $u\leq C+1$ on $\Delta _ j$. Since $x_ j\in B\cap L_ j\subset \Delta _ j$, this gives $u(x_ j)\leq C+1$, contradicting $u(x_ j)\to +\infty $. Hence $u$ is bounded above near every point of $S$.

Step 2: the subharmonic extension. By lemma 5.11.2(1), every compact set $S\cap \overline B$ as above is real polar. Countably many such sets cover $S$, so $S$ is real polar by proposition 4.1.11. By Step 1 and the upper semicontinuity of $u$ on $\Omega \setminus S$, the function $u$ is bounded above on $K\setminus S$ for every compact $K\subset \Omega $. Moreover $u$ is subharmonic on $\Omega \setminus S$ by theorem 5.9.1. By theorem 3.4.27, $u$ has a subharmonic extension $\widetilde u$ to $\Omega $, given on $S$ by the upper-limit formula. Since $S$ is Lebesgue null and $u$ is finite almost everywhere on $\Omega \setminus S$, the function $\widetilde u$ is not identically $-\infty $ on the domain $\Omega $, so it is locally integrable.

Step 3: $\widetilde u$ is plurisubharmonic. We verify the hypothesis of lemma 5.8.2 for the locally integrable function $\widetilde u$. Fix a unit vector $a\in \mathbb {C}^ n$ and a nonnegative $\chi \in C^\infty _ c(\Omega )$, and write $z=v+ta$ with $v\in (\mathbb {C}a)^\perp $ and $t\in \mathbb {C}$, as in that lemma. The compact set $S\cap \operatorname{Supp}\chi $ is $\mathcal H^{2n-3}$-null, since it is covered by finitely many of the sets $S\cap \overline B$ of the first paragraph, so by lemma 5.11.2(3) its orthogonal projection onto $(\mathbb {C}a)^\perp $ is Lebesgue null. For $v$ outside this projection, the line $\{ v+ta:t\in \mathbb {C}\} $ meets $\operatorname{Supp}\chi $ only in $\Omega \setminus S$. Thus the open set $G_ v=\{ t:v+ta\in \Omega \setminus S\} $ contains $K_ v=\{ t:v+ta\in \operatorname{Supp}\chi \} $, and its connected components are the connected components of an affine complex-line section of $\Omega \setminus S$, on each of which the restriction of $\widetilde u=u$ is subharmonic or identically $-\infty $. As $a$ and $\chi $ are arbitrary, lemma 5.8.2 gives $\mathrm{dd}^{\mathrm{c}}\widetilde u\geq 0$. Since $\widetilde u$ is a locally integrable subharmonic function, it is the canonical representative of its class, and it is plurisubharmonic by theorem 5.9.1. Uniqueness follows from the same theorem, since two plurisubharmonic extensions agree outside the Lebesgue null set $S$. $\square$