PomeDefinition 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 $v_-(\omega )>0$.
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