Lemma 13.13.21.2904B26253E2 Fix a closed smooth real $(1,1)$-form $\theta $ on $X$ representing a big cohomology class. Let $f\colon X\rightarrow \mathbb {R}$ be a lower semicontinuous function. Assume that $P_{\theta }(f)\not\equiv \infty $. Then $P_{\theta }(f)$ is locally bounded outside $\operatorname{nK}(\{ \theta \} )$. Set $U\coloneqq \{ P_{\theta }(f)<f\} \setminus \operatorname{nK}(\{ \theta \} )$, then
0D2A1EE39CF8
13.13.22
\begin{equation} \Bigl (\theta |_ U+\mathrm{dd}^{\mathrm{c}}P_{\theta }(f)|_ U\Bigr )^ n=0. \end{equation}
Proof.
First observe that $V_{\theta }+\inf _ X f$ is a candidate in the definition (13.13.19). So
\[ P_{\theta }(f)\geq V_{\theta }+\inf _ X f, \]
and in particular $P_{\theta }(f)\not\equiv -\infty $.
In particular, $P_{\theta }(f)\in \mathrm{PSH}(X,\theta )$. It is therefore bounded from above. By 13.13.8, it is locally bounded away from $\operatorname{nK}(\{ \theta \} )$.
Now fix $x\in U$, and $\epsilon >0$ so that
\[ P_{\theta }(f)(x)<f(x)-\epsilon . \]
Take holomorphic balls $B\Subset B'\Subset U$ containing $x$. After shrinking $B$, we can guarantee that
BB745DB26624
13.13.23
\begin{equation} \left.P_{\theta }(f)\right|_{B}<f(x)-\epsilon ,\quad f|_ B> f(x)-\epsilon . \end{equation}
Let $\varphi \in \mathrm{PSH}(X,\theta )$ be the balayage of $P_{\theta }(f)$ on $B$, as constructed in proposition 5.17.9. By the domination principle proposition 5.14.22, we find that
\[ \varphi |_{B}\leq f(x)-\epsilon <f|_{B}. \]
Since $\varphi =P_{\theta }(f)$ outside $B$, we conclude that
\[ \varphi \leq f\quad \textup{quasi-everywhere}. \]
From this it follows that $\varphi \leq P_{\theta }(f)$, and hence equality holds. But then (13.13.22) holds automatically on $B$ since $\varphi $ is constructed by balayage. Since $x$ and $B$ are arbitrary, we conclude (13.13.22).
$\square$
Pome
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