Corollary 2.9.28644CAD92A580

Proof. If $n=1$, the only affine complex line is $\mathbb C$ itself. Thus $u$ is subharmonic on every component of $D$, hence upper semicontinuous and locally bounded above.

Let $n\geq 2$, and suppose that $u$ is not bounded above on any neighborhood of some point $p\in D$. Then there are points $x_ j\to p$ with $A_ j=u(x_ j)\to \infty $. Choose $R>0$ with $\overline{B(p,2R)}\subset D$; discarding finitely many terms, we may assume that $|x_ j-p|<R$ for all $j$. Then $\overline{B(x_ j,R)}\subset B(p,2R)\subset D$, and theorem 2.9.22 gives Borel probability measures $\mu _ j$ carried by the sets

with $I_{2n-2}(\mu _ j)\leq C_ nR^{-(2n-2)}$ for every $j$. All the sets $E_ j$ lie in the bounded set $B(p,2R)$. By theorem 2.9.27, some affine complex line $L$ meets $E_ j$ for all $j$ in an infinite set $J$. Choose $y_ j\in L\cap E_ j$ for $j\in J$. Passing to a subsequence, we may assume that $y_ j\to y\in L\cap \overline{B(p,2R)}\subset D$. A small closed disc in $L$ centered at $y$ lies in $D$, so for large $j$ the points $y_ j$ lie in the same connected component of $D\cap L$ as $y$. On this component $u$ is subharmonic, hence upper semicontinuous, and therefore $\varlimsup _ ju(y_ j)\leq u(y)<\infty $. This contradicts $u(y_ j)\geq A_ j\to \infty $. $\square$