Corollary 17.5.39.92C6B04981BA Let $\theta $ be a smooth closed real $(1,1)$-form with $\mathrm{PSH}(X,\theta )\neq \varnothing $, and let $f$ be a continuous real function on $X$ with bounded metric Laplacian. Then
\[ \langle (\theta +\mathrm{dd}^{\mathrm{c}}P_\theta (f))^ n\rangle =\mathbf1_{\{ P_\theta (f)=f\} }(\theta +\mathrm{dd}^{\mathrm{c}}f)^ n, \]
where the right side is understood as in theorem 16.20.2. In particular $\langle (\theta +\mathrm{dd}^{\mathrm{c}}V_\theta )^ n\rangle =\mathbf1_{\{ V_\theta =0\} }\theta ^ n$. If $\{ \theta \} $ is not big, both sides of these two identities vanish.
Proof.
The potential $V_\theta +\min _ Xf$ lies below $f$, so $\mathcal A_\theta (f)$ is nonempty, and it is bounded above by $\max _ Xf$. Put $u=P_\theta (f)$. Since $f$ is continuous, $u\leq f$ everywhere by proposition 16.11.4(1),(2). The scalar case of theorem 16.20.2, with all $\theta _ j=\theta $, $u_ j=u$ and $f_ j=f$, gives
\[ \mathbf1_{\{ u=f\} }\langle \theta _ u^ n\rangle =\mathbf1_{\{ u=f\} }(\theta +\mathrm{dd}^{\mathrm{c}}f)^ n, \]
and corollary 17.5.38 shows that $\langle \theta _ u^ n\rangle $ is carried by $\{ u=f\} $. This proves the first identity. For $f=0$ the envelope $P_\theta (0)$ is $V_\theta $, by definition 10.3.7, which gives the second. If $\{ \theta \} $ is not big, the left sides vanish by corollary 17.5.38, and hence so do the right sides.
$\square$
Pome
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