When $\omega $ is a Kähler form, all bounded $\omega $-plurisubharmonic functions have the same Monge–Ampère mass $\int _ X\omega ^ n$. For a general Hermitian form, the mass can depend on the potential, as in example 11.1.12, and the following two numbers measure its range.
Definition 13.14.14.EB45638CBB2C Let $\omega $ be a Hermitian form on $X$. The lower mass and the upper mass of $\omega $ are
\[ \begin{split} v_-(\omega )& = \inf _{v\in \mathrm{PSH}(X,\omega )\cap L^\infty (X)} \int _ X(\omega +\mathrm{dd}^{\mathrm{c}}v)^ n,\\ v_+(\omega )& = \sup _{v\in \mathrm{PSH}(X,\omega )\cap L^\infty (X)} \int _ X(\omega +\mathrm{dd}^{\mathrm{c}}v)^ n. \end{split} \]
The manifold $X$ has the bounded mass property, abbreviated BMP, if $v_+(\omega )<+\infty $. The form $\omega $ has positive lower mass if $v_-(\omega )>0$.
Testing with $v=0$ gives $v_-(\omega )\leq \int _ X\omega ^ n\leq v_+(\omega )$. By the next proposition, the bounded mass property does not depend on the choice of $\omega $. Since the extended measure (13.14.4) is built from products of bounded potentials, the upper mass also controls the extended measures of unbounded potentials (proposition 13.14.16).
Proposition 13.14.15.45F622BE862A The bounded mass property is independent of the Hermitian form. Moreover, the supremum in definition 13.14.14 is unchanged if only smooth potentials are used.
Proof.
For another Hermitian form $\eta $, choose $C>0$ such that $\eta \leq C\omega $. If $v$ is a bounded $\eta $-plurisubharmonic function, then $v/C$ is $\omega $-plurisubharmonic and
\[ 0\leq \eta +\mathrm{dd}^{\mathrm{c}}v \leq C\bigl (\omega +\mathrm{dd}^{\mathrm{c}}(v/C)\bigr ). \]
By positivity of mixed bounded products, $\int _ X(\eta +\mathrm{dd}^{\mathrm{c}}v)^ n\leq C^ n v_+(\omega )$. Interchanging the two forms proves metric independence.
For the last assertion, approximate a bounded potential by theorem 5.32.1. The smooth approximants decrease and remain uniformly bounded, so the continuity of bounded products along decreasing sequences gives convergence of their total masses. The reverse inequality between the two suprema is immediate.
$\square$
Proposition 13.14.16.1342466C71A9 Assume $v_+(\omega )<+\infty $. If $\theta $ is a smooth real $(1,1)$-form with $\theta \leq C\omega $ for some $C>0$, and $u\in \mathrm{PSH}(X,\theta )$, then
\[ \mathrm{MA}^{\mathrm{ext}}_\theta (u)(X)\leq C^ n v_+(\omega ). \]
Proof.
The comparison in theorem 13.14.3, applied with the Hermitian form $C\omega $, gives
\[ \mathbf1_{D_ t}\mathrm{MA}^{\mathrm{ext}}_\theta (u) \leq \mathbf1_{D_ t}(C\omega +\mathrm{dd}^{\mathrm{c}}u_ t)^ n. \]
Since $u_ t/C$ is bounded and $\omega $-plurisubharmonic, the right-hand side has mass at most $C^ n v_+(\omega )$. Let $t\to \infty $.
$\square$
Pome
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