PomeProposition 16.28.55.9BBE1DE257CB If a smooth real $(1,1)$-form $\theta $ is big, every bounded $u\in \mathrm{PSH}(X,\theta )$ has strictly positive Monge–Ampère mass.
16.28.54 Bounded domination in a semipositive background7A794726C8E7
We return to the compact manifold $X$.
The domination principle for Hermitian backgrounds, theorem 14.19.10, extends to a smooth semipositive background $\theta $ provided every bounded $\theta $-plurisubharmonic potential has positive mass. For bounded potentials $u,v$ satisfying the measure inequality below, contact support bounds $\sup _ XP_\theta (bu-(b-1)v)$ from below as the stretching parameter $b>1$ increases. A fixed gap between $u$ and $v$ would then force the $L^1$ norms of these envelopes, normalized by their suprema, to diverge, contradicting compactness.
We call a smooth semipositive form $\theta $ non-collapsing on bounded potentials if
\[ \int _ X\mathrm{MA}_\theta (w)>0 \quad \hbox{for every }w\in \mathrm{PSH}(X,\theta )\cap L^\infty (X). \]
Every Hermitian form has this property by corollary 14.19.11. Bigness also implies it, by the following envelope argument.
Proof. Choose $\rho \in \mathrm{PSH}(X,\theta )$ and $a>0$ with $\theta _\rho \geq a\omega $, normalize $\rho \leq 0$, and put
\[ p=P_{a\omega }(u-\rho ). \]
The obstacle is finite almost everywhere and bounded below, so lemma 5.40.2 shows that $p$ is bounded. The obstacle is quasi-continuous off the pluripolar pole set of $\rho $. The contact theorem, theorem 16.28.5, applied in the Hermitian background $a\omega $, shows that $\mathrm{MA}_{a\omega }(p)$ is carried by $D=\{ p+\rho =u\} $, outside the pole set of $\rho $.
By lemma 5.24.3, $p+\rho \leq u$ everywhere, and
\[ \theta +\mathrm{dd}^{\mathrm{c}}(p+\rho ) =(a\omega +\mathrm{dd}^{\mathrm{c}}p)+(\theta _\rho -a\omega ) \geq a\omega +\mathrm{dd}^{\mathrm{c}}p\geq 0. \]
On common finite truncation levels, positivity of the mixed products and ordered contact give
\[ \mathbf1_ D\mathrm{MA}_{a\omega }(p) \leq \mathbf1_ D\mathrm{MA}_\theta (p+\rho ) \leq \mathbf1_ D\mathrm{MA}_\theta (u). \]
Exhausting by these levels proves the same inequalities for the extended measures. Since $p$ is bounded, its Monge–Ampère mass with respect to the Hermitian form $a\omega $ is positive by corollary 14.19.11. Integrating the inequalities proves the assertion. $\square$
Theorem 16.28.56 (Guedj–Lu).C67CEBABC968 Let $\theta \geq 0$ be non-collapsing on bounded potentials. Suppose that $u,v\in \mathrm{PSH}(X,\theta )\cap L^\infty (X)$ and $0\leq c<1$. If
\[ \mathbf1_{\{ u<v\} }\mathrm{MA}_\theta (u) \leq c\, \mathbf1_{\{ u<v\} }\mathrm{MA}_\theta (v), \]
then $u\geq v$.
Proof. If the conclusion failed, we could choose $a>0$ such that $E=\{ u<v-a\} $ is nonempty. This set has positive volume by lemma 5.24.4. Fix $C$ with $|u|,|v|\leq C$.
For $b>1$ define the bounded envelope
\[ w_ b=P_\theta \bigl (bu-(b-1)v\bigr ),\qquad D_ b=\{ w_ b=bu-(b-1)v\} . \]
Since $\theta \geq 0$, constants below the obstacle are competitors. By the bounded contact theorem, theorem 16.28.4, the measure $\mathrm{MA}_\theta (w_ b)$ is concentrated on $D_ b$. The potential $b^{-1}w_ b+(1-b^{-1})v$ lies below $u$ quasi-everywhere, hence everywhere by lemma 5.24.3, and it equals $u$ on $D_ b$ outside a pluripolar set. Ordered contact and positivity of the mixed terms give
\[ \mathbf1_{D_ b}\left[b^{-n}\mathrm{MA}_\theta (w_ b) +(1-b^{-1})^ n\mathrm{MA}_\theta (v)\right] \leq \mathbf1_{D_ b}\mathrm{MA}_\theta (u). \]
Choose $b$ so large that $(1-b^{-1})^ n>c$. Then the hypothesis forces $\mathrm{MA}_\theta (w_ b)=0$ on $D_ b\cap \{ u<v\} $. Since $\theta $ is non-collapsing, $w_ b$ has positive total mass, so some of this mass is carried by $D_ b\cap \{ u\geq v\} $. At a point of this set outside the exceptional pluripolar set,
\[ w_ b=bu-(b-1)v=u+(b-1)(u-v)\geq -C. \]
Consequently $\sup _ Xw_ b\geq -C$ for all large $b$.
On the other hand, on the fixed set $E$ the obstacle bound gives $w_ b\leq C-(b-1)a$ quasi-everywhere. Hence
\[ \int _ X|w_ b-\sup _ Xw_ b|\, \mathrm{d}V \geq \bigl ((b-1)a-2C\bigr )\operatorname {vol}_{\mathrm{d}V}(E) \]
for all sufficiently large $b$. This contradicts the uniform $L^1$ bound for normalized potentials in theorem 5.25.2. $\square$
The exponential comparison principle follows as in corollary 14.19.12. For unbounded potentials, the stretched envelopes must also be kept in the required singularity class; this is the role of the relative classes introduced below.
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