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13.12 Quasi-continuous functionsAD5CC24EA726

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Let $X$ be a compact Kähler manifold of dimension $n$ and $\theta $ be a closed real smooth $(1,1)$-form on $X$ representing a big cohomology class.

Definition 13.12.1.C96BE5099B6A Let $A\subseteq X$ be a Borel subset. The $\theta $-capacity $\operatorname{Cap}_{\theta }(A)$ of $A$ is defined as

\[ \operatorname{Cap}_{\theta }(A)\coloneqq \sup \left\{ \int _ A\theta _{\varphi }^ n : \varphi \in \mathrm{PSH}(X,\theta ),V_{\theta }-1\leq \varphi \leq V_{\theta } \right\} . \]
Here $V_{\theta }$ is defined in (13.13.7) below. As we will recall below, the non-pluripolar product $\theta _{\varphi }^ n$ is well-defined. The capacity is not very sensitive to the choice of $\theta $:

Theorem 13.12.2.C5C10F2AB708 Let $\theta '$ be another closed real smooth $(1,1)$-form on $X$ representing a big cohomology class. Then there are continuous functions $f,g\colon [0,\infty )\rightarrow [0,\infty )$, with $f(0)=g(0)=0$, such that for any Borel subset $E\subseteq X$, we have

\[ \operatorname{Cap}_{\theta }(E)\leq f\left( \operatorname{Cap}_{\theta '}(E)\right),\quad \operatorname{Cap}_{\theta '}(E)\leq g\left( \operatorname{Cap}_{\theta }(E)\right). \]

Proof. Use the comparison with Kähler capacity with the reference potentials defining $\operatorname{Cap}_\theta$ and $\operatorname{Cap}_{\theta'}$. It gives continuous nondecreasing functions $F,G,F',G'$, all zero at zero, with \[\operatorname{Cap}_\theta\leq F\circ\operatorname{Cap}_\omega,\quad\operatorname{Cap}_\omega\leq G\circ\operatorname{Cap}_\theta,\quad\operatorname{Cap}_{\theta'}\leq F'\circ\operatorname{Cap}_\omega,\quad\operatorname{Cap}_\omega\leq G'\circ\operatorname{Cap}_{\theta'}.\] The reference potentials have positive mass because the classes are big. Set $f=F\circ G'$ and $g=F'\circ G$. Composing the respective inequalities proves the two assertions for every Borel set. The compositions are continuous and vanish at zero, as required.

A more general result is proved in [Lu21]. Similar comparison results hold between $\theta $-capacity and the classical Bedford–Taylor capacity, see [Section 9.2, GZ17] for the proof. As a consequence, we can freely apply the results in [Section 4.2, GZ17], even though capacity has a different meaning there.

Definition 13.12.3.BA13EE505D9B Let $U$ be an open subset of $X$. A function $f\colon U\rightarrow [-\infty ,\infty ]$ is quasi-continuous if it is Borel measurable and for any $\epsilon >0$, there is an open subset $G\subseteq U$ such that

  1. E9A5229B982B

    $\operatorname{Cap}_{\theta }(G)\leq \epsilon $;

  2. EC64A29E7BCB

    $f|_{U\setminus G}$ is real-valued and continuous.

Thanks to theorem 13.12.2, the notion of quasi-continuous functions is independent of the choice of $\theta $. Note that if $f,g\colon U\rightarrow [-\infty ,\infty ]$ are two Borel measurable functions equal quasi-everywhere (see definition 5.47.2), then $f$ is quasi-continuous if and only if $g$ is.

Theorem 13.12.4.9F9E57CC1422 Let $A\subseteq X$ be a Borel set. Then the following are equivalent:

  1. 9860890C10B2

    $A$ is pluripolar;

  2. 0031AD934DC8

    $\operatorname{Cap}_{\theta }(A)=0$.

Proof. By capacity comparison, zero $\theta$-capacity is equivalent to zero Kähler capacity. The comparison with coordinate-ball capacities makes this equivalent to zero local Bedford--Taylor capacity in every member of a finite coordinate cover. The local zero-capacity theorem identifies these zero sets with the pluripolar sets. This proves both implications. For a Borel set, one may first apply the local statement to compact subsets and then use the capacitability and zero-capacity characterization proved with that theorem; a mere measure-zero argument would not suffice.

See [Theorem 4.40, GZ17] for the proof.

Example 13.12.5.472C675FCEA8 A quasi-plurisubharmonic function on an open subset $U\subseteq X$ is always quasi-continuous. See [Theorem 4.20, GZ17] for the proof.

More generally, if $\varphi ,\psi $ are two quasi-plurisubharmonic functions on $U$, then the following function

\[ f(x)\coloneqq \begin{cases} \varphi (x)-\psi (x),& \textup{if }(\varphi \lor \psi )(x)\neq -\infty ;\\ \infty ,& \textup{otherwise} \end{cases} \]

is quasi-continuous.2

Definition 13.12.6.F7490AE48EAC Let $U$ be an open subset of $X$. Let $(f_ i)_{i\in I}$ be a net of Borel measurable functions $f_ i\colon U\rightarrow [-\infty ,\infty ]$, and $f\colon U\rightarrow [-\infty ,\infty ]$ be a Borel measurable function. We say $(f_ i)_{i\in I}$ converges to $f$ in capacity if for any $\delta >0$, we have

\[ \lim _{i\in I} \operatorname{Cap}_{\theta }\Bigl ( \{ f_ i>f+\delta \} \Bigr )=0,\quad \lim _{i\in I} \operatorname{Cap}_{\theta }\Bigl ( \{ f_ i<f-\delta \} \Bigr )=0{}^{\href{#sgpt-a0000011932}{3}}. \]

We sometimes write $f_ i\xrightarrow {\mathrm{C}} f$.

Note that $f$ is not uniquely determined by the net $(f_ i)_{i\in I}$. Thanks to theorem 13.12.2, the notion of convergence in capacity is independent of the choice of $\theta $.

Proposition 13.12.7.D56D6D4A5E94 Let $U$ be an open subset of $X$. Let $(\varphi _ i)_{i\in I}$ be a net in $\mathrm{PSH}(U,\theta )$ and $\varphi \in \mathrm{PSH}(U,\theta )$. Assume one of the following conditions holds:

  1. 11DA1A704FF9

    $(\varphi _ i)_{i\in I}$ is decreasing and $\varphi $ is the limit of the net;

  2. C18393916362

    $(\varphi _ i)_{i\in I}$ is increasing and converges almost everywhere to $\varphi $.

Then $(\varphi _ i)_{i\in I}$ converges to $\varphi $ in capacity.

See [Proposition 4.25, GZ17] for the proof. The reference concerns only the sequence case, but the proof works for nets as well.

Theorem 13.12.8.272876B00528 Let $U$ be an open subset of $X$. Let $(\chi _ i\colon U\rightarrow [0,\infty ))_ i$ be a sequence of quasi-continuous functions. Assume that the sequence is uniformly bounded and converges in capacity to $\chi \colon U\rightarrow [0,\infty )$. For each $j=1,\ldots ,p$, let $(\varphi _ j^ i)_ i$ be uniformly bounded sequences in $\mathrm{PSH}(U,\theta _ j)$ converging in capacity to bounded $\varphi _ j\in \mathrm{PSH}(U,\theta _ j)$. Then

\[ \chi _ i\, \bigwedge _{j=1}^ p \theta _{j,\varphi _ j^ i}\rightharpoonup \chi \, \bigwedge _{j=1}^ p \theta _{j,\varphi _ j} \]

weakly as currents on $U$.

See [Theorem 4.26, GZ17]1.
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