Proposition 13.17.21C00C4C5A656

Proof. The form $\theta +\eta $ is big: adding the positive current $\eta +\mathrm{dd}^{\mathrm{c}}v$ to a strictly positive current $\theta +\mathrm{dd}^{\mathrm{c}}\rho $ gives a strictly positive current $\theta +\eta +\mathrm{dd}^{\mathrm{c}}(\rho +v)$. For the upper volume, add $v$ to each competitor $u'\sim u$ in $\mathrm{PSH}(X,\theta )$. Then $u'+v\sim u+v$, and positivity of the mixed products shows that $\int _ X\mathrm{MA}_{\theta +\eta }(u'+v)\geq \int _ X\mathrm{MA}_\theta (u')$.

For the lower volume, choose $w\in \mathrm{PSH}(X,\theta +\eta )$ with $w\sim u+v$, and let $p=P_\theta (w-v)$. Since $w-v=u+O(1)$ outside the pole sets, a translate of $u$ is a competitor and $p\leq u+C$ for a constant $C$. Thus $p\in \mathrm{PSH}(X,\theta )$ and $p\sim u$. By the contact theorem, $\mathrm{MA}_\theta (p)$ is carried by $D=\{ p+v=w\} $.

Furthermore $p+v\leq w$ everywhere by lemma 5.24.3. Positivity of the mixed products and ordered contact therefore give

Integrating and taking the infimum over $w$ proves the assertion. $\square$