PomeDefinition 5.47.14.09A00D4B8096 The plurifine topology on $X$ is the topology with a base consisting of sets of the form $F^{-1}(V)$, where $U\subseteq X$ is an open subset, $F\colon U\rightarrow \Omega $ is a biholomorphic map, $\Omega \subseteq \mathbb {C}^ n$ is a domain for some $n\in \mathbb {N}$, and $V\subseteq \Omega $ is $\mathcal{F}$-open.
5.15.17 Plurifine topology on manifolds2403FAC7083C
Proposition 5.15.18.E49CF7960E4C Let $\varphi \in \mathrm{QPSH}(X)$. Then $\varphi $ is $\mathcal{F}$-continuous.
Proof. It suffices to show that $\varphi |_{\{ \varphi \neq -\infty \} }$ is $\mathcal{F}$-continuous, since $\{ \varphi =-\infty \} $ is $\mathcal{F}$-closed.
The problem is local, so we may assume that $X\subseteq \mathbb {C}^ n$ is a domain and $\varphi =\psi +g$, where $\psi \in \mathrm{PSH}(X)$ and $g\in C^{\infty }(X)$ and $|g|\leq C$ for some $C>0$. For an open interval $(a,b)\subseteq \mathbb {R}$, it suffices to show that
\[ U\coloneqq \left\{ x\in X:a<\varphi (x)<b\right\} =\left\{ x\in X:a-g(x)<\psi (x)<b-g(x)\right\} \]
is $\mathcal{F}$-open. Take $x\in U$. We can find an open neighborhood $V$ of $x$ in $U$ such that
\[ \sup _{y\in V} \left( a-g(y)\right)<\psi (x)< \inf _{y\in V} \left( b-g(y) \right). \]
Therefore,
\[ \left\{ z\in V:\sup _{y\in V} \left(a-g(y)\right)<\psi (z)<\inf _{y\in V} \left(b-g(y)\right)\right\} \]
is an $\mathcal{F}$-open neighborhood of $x$ in $U$. We conclude that $U$ is $\mathcal{F}$-open. $\square$
Corollary 5.15.19.030D740FC9C6 Let $\varphi ,\psi \in \mathrm{QPSH}(X)$. Then the set
\[ \{ x\in X: \varphi (x)>\psi (x)\} \]
is $\mathcal{F}$-open.
Proof. It suffices to show that for any $x\in X$ such that $\varphi (x)>\psi (x)$, the same holds on an $\mathcal{F}$-neighborhood $U$ of $x$. Observe that $\varphi (x)\neq -\infty $. If $\psi (x)\neq -\infty $, then it suffices to apply proposition 5.15.18. Otherwise, take
\[ U\coloneqq \{ y\in X: \varphi (y)>\varphi (x)-1\} \cap \{ y\in X: \psi (y)<\varphi (x)-1\} . \]
This is an $\mathcal{F}$-open neighborhood of $x$ contained in $\{ \varphi >\psi \} $. $\square$
Lemma 5.15.20.4014761D7A8A Let $Z\subseteq X$ be a pluripolar subset. Then
\[ \overline{X\setminus Z}^{\mathcal{F}}=X. \]
Proof. The problem is local, so we may assume that $X$ is a domain in $\mathbb {C}^ n$ and that there is $\varphi \in \mathrm{PSH}(X)$ with $Z\subseteq \{ \varphi =-\infty \} $. It suffices to show that $\{ \varphi >-\infty \} $ is $\mathcal{F}$-dense.
Let $x\in X$ be a point with $\varphi (x)=-\infty $ and $U\subseteq X$ be an $\mathcal{F}$-open neighborhood of $x$ in $X$. We need to show that $U\cap \{ \varphi >-\infty \} \neq \varnothing $.
Thanks to theorem 5.15.11, after shrinking $U$, we may assume that there is $\psi \in \mathrm{PSH}(X)$ such that $U=\{ \psi >0\} $. Observe that $U$ is not a pluripolar set: Otherwise, $\psi \leq 0$ almost everywhere by proposition 5.14.4, and hence everywhere by proposition 5.14.6. So $\varphi |_ U\not\equiv -\infty $. We conclude that $U\cap \{ \varphi >-\infty \} \neq \varnothing $, as desired. $\square$
Corollary 5.15.21.31032260EF62 Let $\varphi ,\psi \in \mathrm{QPSH}(X)$. Set
\[ W=\left\{ x\in X:\varphi (x)=-\infty \right\} \textup{ or }W=\left\{ x\in X:\psi (x)=-\infty \right\} . \]
Then for any pluripolar set $Z\subseteq X$, we have
\[ \sup _{X\setminus W}(\varphi -\psi ) =\sup _{X\setminus (W\cup Z)}(\varphi -\psi ),\quad \inf _{X\setminus W}(\varphi -\psi ) =\inf _{X\setminus (W\cup Z)}(\varphi -\psi ). \]
In particular, taking $\psi =0$, we find that
\[ \sup _{X\setminus Z}\varphi =\sup _ X \varphi . \]
Proof. This is an immediate consequence of 5.15.20 and proposition 5.15.18. $\square$
In the literature about pluripotential theory, one often finds the careless expressions like $\sup _ X (\varphi -\psi )$. The issue is that $\varphi -\psi $ is not defined everywhere, and hence this expression does not make sense if you read it literally. corollary 5.15.21 tells you that you do not have to worry too much about the details on a pluripolar set. In other words, $\sup $ and $\inf $ could always be understood as a kind of essential supremum and essential infimum modulo pluripolar sets.
There is a convenient way to fix this issue in the literature — just replace the suprema by quasi-suprema:Definition 5.15.22.8485EC5E3BC1 Let $Z\subseteq X$ be a pluripolar set, and $f\colon X\setminus Z\rightarrow [-\infty ,\infty ]$ be a function. We define the quasi-supremum and the quasi-infimum of $f$ as follows:
\[ \begin{split} \operatorname *{q-sup}_ X f\coloneqq & \inf \left\{ \sup _{X\setminus W} f:W\supseteq Z, W \textup{ is pluripolar}\right\} ,\\ \operatorname *{q-inf}_ X f\coloneqq & \sup \left\{ \inf _{X\setminus W} f:W\supseteq Z, W \textup{ is pluripolar}\right\} . \end{split} \]
For a quasi-plurisubharmonic function $\varphi $, we have
\[ \operatorname *{q-sup}_ X \varphi =\sup _ X \varphi , \]
as a consequence of corollary 5.15.21.
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