Proposition 16.5.5 (Comparison principle).FDFD0A6F6287 Fix $j\in \{ 0,\ldots ,n\} $. Let $\theta _1,\ldots ,\theta _ j$ be closed smooth real $(1,1)$-forms on $X$ and $\psi _ i\in \mathrm{PSH}(X,\theta _ i)$ for $i=1,\ldots ,j$. Suppose that $\varphi ,\psi \in \mathcal{E}(X,\theta ;\phi )$. Then
EAAF267DE59E
16.5.6
\begin{equation} \int _{\{ \varphi <\psi \} } \theta _{\varphi }^{n-j} \wedge \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{j,\psi _ j}\geq \int _{\{ \varphi <\psi \} } \theta _{\psi }^{n-j} \wedge \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{j,\psi _ j}. \end{equation}
As a consequence, we also have
D3168D446564
16.5.7
\begin{equation} \int _{\{ \varphi \leq \psi \} } \theta _{\varphi }^{n-j} \wedge \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{j,\psi _ j}\geq \int _{\{ \varphi \leq \psi \} } \theta _{\psi }^{n-j} \wedge \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{j,\psi _ j}. \end{equation}
Proof.
Observe that $\varphi \lor \psi \in \mathcal{E}(X,\theta ;\phi )$, as a consequence of the monotonicity theorem theorem 13.13.11. Using the plurifine locality of the non-pluripolar product proposition 13.11.4 1, we compute
\[ \begin{aligned} & \int _ X \theta _{\varphi \lor \psi }^{n-j}\wedge \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{j,\psi _ j}\\ \geq & \int _{\{ \varphi >\psi \} } \theta _{\varphi }^{n-j}\wedge \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{j,\psi _ j}+\int _{\{ \varphi <\psi \} } \theta _{\psi }^{n-j}\wedge \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{j,\psi _ j}\\ =& \int _ X \theta _{\varphi }^{n-j}\wedge \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{j,\psi _ j}-\int _{\{ \varphi \leq \psi \} } \theta _{\varphi }^{n-j}\wedge \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{j,\psi _ j}\\ & +\int _{\{ \varphi <\psi \} } \theta _{\psi }^{n-j}\wedge \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{j,\psi _ j}\\ =& \int _ X \theta _{\varphi \lor \psi }^{n-j}\wedge \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{j,\psi _ j}-\int _{\{ \varphi \leq \psi \} } \theta _{\varphi }^{n-j}\wedge \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{j,\psi _ j}\\ & +\int _{\{ \varphi <\psi \} } \theta _{\psi }^{n-j}\wedge \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{j,\psi _ j}, \end{aligned} \]
where in the last step, we applied proposition 16.3.10 to $\varphi $ and $\varphi \lor \psi $. Therefore,
\[ \begin{aligned} \int _{\{ \varphi \leq \psi \} }& \theta _{\varphi }^{n-j} \wedge \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{j,\psi _ j}\\ & \geq \int _{\{ \varphi <\psi \} } \theta _{\psi }^{n-j} \wedge \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{j,\psi _ j}. \end{aligned} \]
Applying the preceding inequality to $\varphi +\epsilon $ in place of $\varphi $, and using $\theta _{\varphi +\epsilon }=\theta _\varphi $, we get
\[ \int _{\{ \varphi +\epsilon \leq \psi \} }\theta _\varphi ^{n-j}\wedge \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{j,\psi _ j} \geq \int _{\{ \varphi +\epsilon <\psi \} }\theta _\psi ^{n-j}\wedge \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{j,\psi _ j}. \]
As $\epsilon \searrow 0$, by monotone convergence theorem, we obtain
\[ \int _{\{ \varphi <\psi \} }\theta _\varphi ^{n-j}\wedge \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{j,\psi _ j} \geq \int _{\{ \varphi <\psi \} }\theta _\psi ^{n-j}\wedge \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{j,\psi _ j}, \]
which proves (16.5.6).
Finally, the (16.5.7) follows by subtracting (16.5.6) (with the role of $\varphi $ and $\psi $ switched) from the total mixed mass equality
\[ \int _ X\theta _\varphi ^{n-j}\wedge \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{j,\psi _ j}= \int _ X\theta _\psi ^{n-j}\wedge \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{j,\psi _ j}, \]
which itself follows from proposition 16.3.10 and proposition 16.5.4.
$\square$
Pome
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