The classical fine topology of definition 4.2.2 is the coarsest topology that contains the Euclidean topology and makes all subharmonic functions continuous. Replacing subharmonic by plurisubharmonic functions, we obtain the plurifine topology, whose terminology below parallels that of subsection 4.2.1. On an open subset of $\mathbb C^ n=\mathbb R^{2n}$ every plurisubharmonic function is subharmonic (theorem 5.9.1), so every plurifine open set is finely open. The main results of this section, a base of plurifine neighborhoods and the positive volume of plurifine open sets, are the plurifine counterparts of proposition 4.2.4, proposition 4.2.5.
For $n=1$ the two topologies coincide. Indeed, let $V\subset \mathbb C$ be open, let $u\in \mathrm{SH}(V)$, and let $V'$ be the union of the connected components of $V$ on which $u$ is not identically $-\infty $. Then $u|_{V'}$ is a plurisubharmonic function on $V'$, and $u=-\infty $ on $V\setminus V'$; hence for every $t\in \mathbb R$ the superlevel set $\{ u>t\} $ is a superlevel set of $u|_{V'}$, which is plurifine open, while the sublevel sets $\{ u<s\} $ are Euclidean open. Thus every subharmonic function is continuous for the plurifine topology, and every finely open set is plurifine open. For $n\ge 2$ the plurifine topology is strictly coarser (proposition 4.2.25, example 5.47.17).
Definition 5.47.14.09A00D4B8096 Let $X$ be a complex manifold. The plurifine topology on $X$ is the coarsest topology on $X$ that contains the topology of $X$, which we call the Euclidean topology, and makes every $u\in \mathrm{PSH}(V)$, for every open set $V\subset X$, continuous as a map from $V$ with the induced topology to $[-\infty ,\infty )$. Sets that are open in this topology are called plurifine open, and neighborhoods in this topology are called plurifine neighborhoods. A set $E\subset X$ is plurithin at a point $a\in X$ if either $a$ does not belong to the Euclidean closure of $E\setminus \{ a\} $, or there is a plurisubharmonic function $\psi $ on a Euclidean open neighborhood of $a$ such that $\psi (a)$ is finite and
\[ \varlimsup _{E\setminus \{ a\} \ni z\to a}\psi (z)<\psi (a), \]
where the upper limit is taken with respect to the Euclidean topology.
Since the sublevel sets $\{ u<s\} $ of upper semicontinuous functions are Euclidean open, the plurifine topology has a subbasis consisting of the Euclidean open sets and the sets $\{ z\in V:u(z)>t\} $, where $V\subset X$ is open, $u\in \mathrm{PSH}(V)$ and $t\in \mathbb R$. Because $\{ u>t\} =\{ \max \{ u,t-1\} >t\} $, it suffices to use locally bounded functions $u$ here.
On $\mathbb C^ n$ one may even use finite plurisubharmonic functions defined on all of $\mathbb C^ n$. Indeed, let $u$ be plurisubharmonic on a neighborhood of a closed ball $\overline{B(a,r)}$, with $m\le u\le M$ on $B(a,r)$. Choose $A>0$ so large that the function $q(z)=m-1+A(|z-a|^2-r^2/16)$ exceeds $M$ where $|z-a|\ge r/2$, and note that $q<m$ on $B(a,r/4)$. By proposition 5.9.5, the function equal to $\max \{ u,q\} $ on $B(a,r)$ and to $q$ outside $B(a,r)$ is plurisubharmonic on $\mathbb C^ n$; it is finite and coincides with $u$ on $B(a,r/4)$. Hence, for every $t\in \mathbb R$, the set $B(a,r/4)\cap \{ u>t\} $ is the intersection of a ball with a superlevel set of a finite plurisubharmonic function on $\mathbb C^ n$, and every subbasic set $\{ z\in V:u(z)>t\} $ with locally bounded $u$ is a union of such sets. Since, moreover, the functions $\pm \operatorname{Re}z_ j$ and $\pm \operatorname{Im}z_ j$ are finite and plurisubharmonic on $\mathbb C^ n$ and generate the Euclidean topology, the plurifine topology of $\mathbb C^ n$ is the coarsest topology that makes all finite plurisubharmonic functions on $\mathbb C^ n$ continuous.
The superlevel sets $\{ u>t\} $ need not be Euclidean open. For example, let $D=\{ z\in \mathbb C^ n:|z_1|<1/2\} $, put $a_ j=\mathrm e^{-j}$ for $j\ge 1$, and let $u=\sum _{j\ge 1}2^{-j}\log |z_1-a_ j|$ on $D$. Each term is a positive multiple of the function $\log |z_1-a_ j|^2$, which is plurisubharmonic on the domain $D$ by proposition 5.10.3, and each term is negative on $D$, because $|z_1-a_ j|<1/2+\mathrm e^{-1}<1$. Hence the partial sums form a decreasing sequence of plurisubharmonic functions (proposition 5.9.5), and since $u(0)=-\sum _ jj2^{-j}=-2$ is finite, $u\in \mathrm{PSH}(D)$ by theorem 5.10.1. The point $0$ lies in the plurifine open set $\{ u>-3\} $, whereas the points $(a_ j,0,\ldots ,0)$, at which $u=-\infty $, converge to $0$. Thus $\{ u>-3\} $ is not a Euclidean neighborhood of $0$.
Proposition 5.47.15.2297E8BC91FF Let $X$ be a complex manifold and $a\in X$.
A set $U\subset X$ containing $a$ is a plurifine neighborhood of $a$ if and only if $X\setminus U$ is plurithin at $a$.
In a chart around $a$, the sets $B\cap \{ \psi >0\} $ containing $a$, where $B$ is a sufficiently small coordinate ball centered at $a$ and $\psi $ is a bounded plurisubharmonic function on a neighborhood of $\overline B$, form a base of plurifine neighborhoods of $a$.
The plurifine topology of an open subset of $X$ is the topology induced by the plurifine topology of $X$. Biholomorphic maps are homeomorphisms for the plurifine topologies, and holomorphic maps between complex manifolds are continuous for the plurifine topologies.
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
\[ N_0=W_0\cap \{ u_1>c_1\} \cap \cdots \cap \{ u_ k>c_ k\} \]
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
\[ \psi =\frac14+\sum _{i=1}^ k\frac{u_ i-u_ i(a)}{\delta _ i}. \]
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.
$\square$
Proposition 5.47.16.DD1F66C6842A Let $X$ be a complex manifold.
Every nonempty plurifine open subset of $X$ contains a Borel set of positive Lebesgue measure in some chart.
The complement of every pluripolar set is dense for the plurifine topology.
For every $u\in \mathrm{PSH}(X)$, the set $\{ u>-\infty \} $ is plurifine open.
Proof.
(1) Let $O$ be a nonempty plurifine open set and $a\in O$. By proposition 5.47.15(2), $O$ contains a set $A=B\cap \{ \psi >0\} $ with $a\in A$, where $B$ is a coordinate ball and $\psi $ is a bounded plurisubharmonic function on a neighborhood of $\overline B$. This set is Borel because $\psi $ is upper semicontinuous. If $A$ had measure zero, then $\max \{ \psi ,0\} =0$ almost everywhere on $B$; since both functions are plurisubharmonic on $B$, they would coincide everywhere by the uniqueness in theorem 5.9.1, contradicting $\psi (a)>0$.
(2) A pluripolar set is Lebesgue null in every chart, so by (1) it contains no nonempty plurifine open set. Equivalently, its complement meets every nonempty plurifine open set.
(3) The set $\{ u>-\infty \} =\bigcup _ m\{ u>-m\} $ is a union of subbasic plurifine open sets.
$\square$
Example 5.47.17.7E1189000F74 Let $n\ge 2$ and $H=\{ z\in \mathbb C^ n:z_1=0\} $. Then $H$ is pluripolar, being the pole set of $\log |z_1|^2$ (proposition 5.10.3), but no point of $H$ is isolated in the topology induced on $H$ by the plurifine topology. This is the second assertion of proposition 4.2.25, because the topology $\tau $ of that proposition is the plurifine topology. Indeed, every $u\in \mathrm{PSH}(V)$, with $V\subset \mathbb C^ n$ open, satisfies the conditions in the definition of $\tau $, so every plurifine open set is open for $\tau $. Conversely, let $u\colon V\to [-\infty ,\infty )$ be as in the definition of $\tau $. By definition 5.8.1, $u$ is plurisubharmonic on the union $V'$ of the connected components of $V$ on which it is not identically $-\infty $. Hence its superlevel sets $\{ u>t\} =\{ z\in V':u(z)>t\} $, $t\in \mathbb R$, are plurifine open, while its sublevel sets are Euclidean open. Thus $u$ is continuous for the plurifine topology, and every $\tau $-open set is plurifine open.
Consequently, neither conclusion of proposition 4.2.6 carries over to pluripolar sets: $H$ is not plurifinely discrete, and for $a\in H$ the pluripolar set $H\setminus \{ a\} $ is not plurifine closed, since every plurifine neighborhood of $a$ meets it.
Example 5.47.18.02396DC16057 For every $n\ge 1$ there is a plurifine open subset of $\mathbb C^ n$ that is not a Borel set.
Let $K\subset [0,1/10]\subset \mathbb C$ be the compact polar set and $\mu $ the atom-free probability measure carried by $K$ of proposition 4.1.12, and put $F(x)=\mu (K\cap (-\infty ,x])$ for $x\in \mathbb R$. The function $F$ is nondecreasing, $F=0$ on $(-\infty ,0)$ and $F=1$ on $[1/10,\infty )$. It is continuous: it is right continuous because $\mu $ is finite, and its jump at $x$ is $\mu (K\cap \{ x\} )=0$, because $\mu $ has no atoms. For $y\in (0,1)$ the set $\{ x\in \mathbb R:F(x)\ge y\} $ is closed, nonempty and bounded below, and we let $g(y)$ be its smallest element. Then $F(g(y))\ge y$, and $F(x)<y$ for $x<g(y)$, so $F(g(y))=y$ because $F$ is continuous at $g(y)$. Hence $g$ is injective; it is also nondecreasing, so $g$ is strictly increasing and in particular Borel measurable. Moreover $g(y)\in K$. Otherwise, since $K$ is closed, there is $\delta >0$ with $[g(y)-\delta ,g(y)]\cap K=\varnothing $; then $F(g(y)-\delta )=F(g(y))=y$, which contradicts the minimality of $g(y)$.
Let $N\subset (0,1)$ be a set that is not Lebesgue measurable, and put $A=g(N)\subset K$. Since $g$ is injective, $N=g^{-1}(A)$; so if $A$ were a Borel subset of $\mathbb C$, then $A=A\cap \mathbb R$ would be a Borel subset of $\mathbb R$, and $N$ would be a Borel set. Hence $A$ is not a Borel subset of $\mathbb C$. As a subset of the polar set $K$, the set $A$ is polar, hence finely closed in $\mathbb C=\mathbb R^2$ by proposition 4.2.6. Since the fine and the plurifine topologies of $\mathbb C$ coincide, as shown at the beginning of this section, $W_1=\mathbb C\setminus A$ is a plurifine open subset of $\mathbb C$ which is not Borel. For $n\ge 2$, the set $W=W_1\times \mathbb C^{n-1}$ is the preimage of $W_1$ under the holomorphic projection $z\mapsto z_1$, so it is plurifine open by proposition 5.47.15(3). It is not Borel, because $W_1$ is its preimage under the continuous map $\zeta \mapsto (\zeta ,0,\ldots ,0)$.
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