Pome32986EE9790232986EE97902
Choose a Hermitian form $\Omega \geq \theta $. By proposition 13.14.16, theorem 13.14.3, the mass of every $\theta $-plurisubharmonic potential is at most $v_+(\Omega )$. Hence, for $u\in \mathrm{PSH}(X,\theta )$,
If $\theta $ is closed, both volumes equal $\int _ X\mathrm{MA}_\theta (u)$, because in the closed theory the non-pluripolar mass depends only on the singularity type Theorem 13.8.7. For a nonclosed background the two volumes can differ even at $u=0$, as example 11.1.12 shows.
Both volumes increase when the background increases or the singularities become milder. In the proof, a rooftop envelope for the lower volume and a maximum for the upper volume produce competitors of the required singularity types, and locality of non-pluripolar products confines the error to sets whose mass tends to zero.
Theorem 13.17.1.D7019DA65027 Let $\theta _1,\theta _2$ be smooth real $(1,1)$-forms on $X$. Suppose $\theta _1$ is big, $\theta _1\leq \theta _2$, and $u_ i\in \mathrm{PSH}(X,\theta _ i)$ satisfy $u_1\preceq u_2$. Then
Proof. We first treat the lower volume. After adding a constant to $u_1$ we may assume $u_1\leq u_2$. Choose $v_2\in \mathrm{PSH}(X,\theta _2)$ with $v_2\sim u_2$ and, for real $t>0$, consider the rooftop envelope
Since $u_1\leq u_2\leq v_2+C$ for a constant $C$, the function $u_1-C$ is a competitor; moreover $p_ t\leq u_1+t$. Hence $p_ t\sim u_1$. By the contact theorem (theorem 15.8.6), the measure $\mathrm{MA}_{\theta _1}(p_ t)$ is carried by the union of $\{ p_ t=u_1+t\} \subset \{ u_1\leq v_2-t\} $ and $\{ p_ t=v_2\} $. On the first set, ordered contact in background $\theta _1$ (proposition 13.14.6) bounds it by $\mathrm{MA}_{\theta _1}(u_1)$. On the second set, $p_ t$ is also $\theta _2$-plurisubharmonic and lies below $v_2$; positivity of the mixed products (lemma 13.14.2) and ordered contact in background $\theta _2$ bound it by $\mathrm{MA}_{\theta _2}(v_2)$. Therefore
As $t\to +\infty $, the first measure on the right has total mass tending to zero: the sets $\{ u_1\leq v_2-t\} $ decrease to a subset of the pole set of $u_1$, and $\mathrm{MA}_{\theta _1}(u_1)$ is a finite measure that puts no mass on pluripolar sets. Integrating and letting $t\to +\infty $ gives $\underline{\operatorname {vol}}_{\theta _1}(u_1)\leq \int _ X\mathrm{MA}_{\theta _2}(v_2)$, and taking the infimum over $v_2$ proves the first inequality.
For the upper volume, choose any $v_1\in \mathrm{PSH}(X,\theta _1)$ with $v_1\sim u_1$, and put $w_ t=\max (v_1,u_2-t)$ for $t>0$. This function is $\theta _2$-plurisubharmonic and satisfies $w_ t\sim u_2$. On the set $D_ t=\{ v_1>u_2-t\} $, locality and positivity of the mixed products imply
The sets $D_ t$ exhaust the complement of the pole set of $v_1$, which carries no mass. Integrating and letting $t\to +\infty $ therefore gives $\int _ X\mathrm{MA}_{\theta _1}(v_1)\leq \overline{\operatorname {vol}}_{\theta _2}(u_2)$, and taking the supremum over $v_1$ proves the second inequality. $\square$
Adding a positive current also increases both volumes. For the lower volume, the potential of the added current is subtracted inside an envelope, which produces a competitor in the original background.
Proposition 13.17.2.1C00C4C5A656 Let $u\in \mathrm{PSH}(X,\theta )$ and $v\in \mathrm{PSH}(X,\eta )$ for a smooth real $(1,1)$-form $\eta $ on $X$. Then, for either the lower or the upper volume,
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. $\square$
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