Proposition 5.47.152297E8BC91FF

Proof. We begin with the first two assertions of (3). Let $W\subset X$ be open. Every subbasic set of the plurifine topology of $W$ is a subbasic set of the plurifine topology of $X$. Conversely, the intersection of a subbasic set of $X$ with $W$ is subbasic for $W$, because the restriction of a function $u\in \mathrm{PSH}(V)$ to $V\cap W$ is plurisubharmonic. Since the sets $S\cap W$, with $S$ subbasic for $X$, form a subbasis of the induced topology, the plurifine topology of $W$ is the induced topology.

Next, if $F$ is a biholomorphism, then $F^{-1}(\{ u>t\} )=\{ u\circ F>t\} $, and $u\circ F$ is plurisubharmonic whenever $u$ is. Indeed, by proposition 5.9.3, in connected charts $u\circ F$ is plurisubharmonic or identically $-\infty $, and the second alternative would make $u$ identically $-\infty $ on a nonempty open set, which is impossible because $u$ is locally integrable (theorem 5.9.1). Applying this to $F$ and to $F^{-1}$, we see that both map subbasic sets to subbasic sets. In particular, we may compute plurifine neighborhoods of $a$ in a chart, as if $X$ were an open subset of $\mathbb C^ n$.

(2) Each set $B\cap \{ \psi >0\} $ as in (2) is plurifine open, as the intersection of a Euclidean open set with a subbasic set. Now let $N$ be a plurifine neighborhood of $a$. By the definition of the topology, $N$ contains a set

containing $a$, where $W_0$ is a Euclidean open neighborhood of $a$, $k\ge 0$, each $u_ i$ is plurisubharmonic on a Euclidean open neighborhood $V_ i$ of $a$, and $c_ i\in \mathbb R$. If $k=0$, every coordinate ball $B\subset W_0$ centered at $a$ works, with $\psi =1$. Let $k\ge 1$. Replacing $u_ i$ by $\max \{ u_ i,c_ i-1\} $, which does not change $\{ u_ i>c_ i\} $, we may assume that the $u_ i$ are locally bounded. Put $\delta _ i=u_ i(a)-c_ i>0$. By upper semicontinuity there is a coordinate ball $B'$ centered at $a$, contained in $W_0$ and in every $V_ i$, on which every $u_ i$ is bounded and satisfies $u_ i<u_ i(a)+\delta _ i/(2k)$. Let $B$ be any coordinate ball centered at $a$ with $\overline B\subset B'$, and put

This is a bounded plurisubharmonic function on $B'$ with $\psi (a)=1/4>0$. If $z\in B$ and $u_ i(z)\le c_ i$ for some $i$, then the $i$-th term of the sum is at most $-1$ and each of the other $k-1$ terms is less than $1/(2k)$, so $\psi (z)\le 1/4-1+(k-1)/(2k)<0$. Hence $B\cap \{ \psi >0\} \subset N_0\subset N$, which proves (2).

(1) Let $U$ be a plurifine neighborhood of $a$, and put $E=X\setminus U$; note that $a\notin E$. By (2), $U$ contains a set $B\cap \{ \psi >0\} $ with $\psi (a)>0$, so $\psi \le 0$ on $E\cap B$. Hence either $a$ is not in the Euclidean closure of $E$, or $\varlimsup _{E\ni z\to a}\psi (z)\le 0<\psi (a)$; in both cases $E$ is plurithin at $a$. Conversely, suppose that $E=X\setminus U$ is plurithin at $a\in U$. If $a$ is not in the Euclidean closure of $E$, then $U$ contains a Euclidean neighborhood of $a$. Otherwise let $\psi $ be as in definition 5.47.14, and choose $c$ with $\varlimsup _{E\ni z\to a}\psi (z)<c<\psi (a)$. Then $\psi <c$ on $E\cap B$ for a small ball $B$ centered at $a$, so the plurifine open set $B\cap \{ \psi >c\} $ contains $a$ and is contained in $U$.

(3) It remains to prove that a holomorphic map $F\colon Y\to X$ is continuous for the plurifine topologies. Preimages of Euclidean open sets are Euclidean open. Let $u\in \mathrm{PSH}(V)$ with $V\subset X$ open, let $t\in \mathbb R$, and let $y\in F^{-1}(\{ u>t\} )$. Choose a connected coordinate neighborhood $Y'$ of $y$ such that $F(Y')$ lies in a connected coordinate neighborhood contained in $V$. By proposition 5.9.3, $u\circ F$ is either identically $-\infty $ or plurisubharmonic on $Y'$; the first alternative is excluded because $u(F(y))>t$. Thus $Y'\cap \{ u\circ F>t\} $ is a plurifine open neighborhood of $y$ contained in $F^{-1}(\{ u>t\} )$. Hence preimages of subbasic sets are plurifine open. $\square$