Theorem 16.23.5641CB44C78E9

Proof. We first reduce to model potentials, and then apply contact concentration to the rooftop envelopes of their truncations.

If $u$ has mass zero, the inequality follows from the monotonicity of mass, $\int _ X\langle \theta _ v^ n\rangle \leq \int _ X\langle \theta _{\max \{ u,v\} }^ n\rangle $, and similarly if $v$ has mass zero. Otherwise, subtracting the same constant from $u$ and $v$, which subtracts it from $\max \{ u,v\} $ and $w$ and changes no mass, we may assume that $u,v\leq 0$. Put $\phi =P_\theta [u]$ and $\psi =P_\theta [v]$; by theorem 16.22.11, these are model potentials with the same masses as $u$ and $v$, respectively, and $u\leq \phi $, $v\leq \psi $, since $u$ and $v$ are competitors for their own singularity envelopes. Put also $\gamma =\max \{ \phi ,\psi \} $ and $r=P_\theta (\phi ,\psi )$.

The four masses in the inequality do not change when $u,v$ are replaced by $\phi ,\psi $. Indeed, for every integer $C$ the potential $\max \{ P_\theta (u+C,V_\theta ),P_\theta (v+C,V_\theta )\} $ has the singularity type of $\max \{ u,v\} $, by proposition 16.11.6, and by proposition 16.21.2(2) these potentials increase almost everywhere to $\gamma $ as $C\to \infty $; so $\gamma $ has the same mass as $\max \{ u,v\} $, by corollary 13.8.9. Moreover $w$ is a common minorant of $\phi $ and $\psi $, so $r\geq w$ is a potential. If $r$ has mass zero, then so has $w$, by monotonicity. If the mass of $r$ is positive, then $w$ has the same mass as $r$ by proposition 16.23.3, applied to $u\in \mathcal E(X,\theta ,\phi )$ and $v\in \mathcal E(X,\theta ,\psi )$. It therefore suffices to prove the theorem for the model potentials $\phi $ and $\psi $, whose maximum is $\gamma $ and whose rooftop envelope is $r$.

Put $m=\int _ X\langle \theta _\gamma ^ n\rangle $, and for $C>0$ let

These potentials have the singularity type of $\gamma $, hence mass $m$, and $r_ C$ decreases to $r$ as $C\to \infty $. By proposition 16.21.3, the measure $\mu _\phi $ is carried by $\{ \phi =0\} $, which is contained in the plurifine open set $\{ \phi >\gamma -C\} $, where $\phi _ C=\phi $. Plurifine locality therefore gives

and the last summand has mass $m-\int _ X\langle \theta _\phi ^ n\rangle $; the analogous decomposition holds for $\psi _ C$. By corollary 16.13.5, $\mu _{r_ C}\leq \mathbf1_{\{ r_ C=\phi _ C\} }\mu _{\phi _ C}+\mathbf1_{\{ r_ C=\psi _ C\} }\mu _{\psi _ C}$. On the set $\{ r_ C<0\} $, the summand $\mu _\phi $ of $\mu _{\phi _ C}$ contributes nothing to this bound, because at a point of $\{ \phi =0\} \cap \{ r_ C=\phi _ C\} $ we have $r_ C=\phi _ C=0$; the same holds for $\mu _\psi $. Consequently

Since $r_ C\leq 0$ and $\int _ X\langle \theta _{r_ C}^ n\rangle =m$, it follows that

Applying lemma 16.23.4 along the integers $C\to \infty $ to the potentials $r_ C$, which decrease to $r$, we find that

which is the assertion for $\phi $ and $\psi $. $\square$