PomeCorollary 10.31.23.C8ACE6D27FAB Suppose that $X$ has the bounded mass property and $v_-(\omega )>0$. Then $\underline{\operatorname {vol}}_\theta (V_\theta )>0$ for every smooth big real $(1,1)$-form $\theta $ on $X$; hence the preceding theorem applies to every such $\theta $.
Corollary 10.31.23C8ACE6D27FAB
Proof. Choose $\rho \in \mathrm{PSH}(X,\theta )$ with $\theta _\rho \geq a\omega $ for some $a>0$. Then $\rho \in \mathrm{PSH}(X,\theta -a\omega )$. Volume addition (proposition 13.17.2), followed by monotonicity in the singularities (theorem 13.17.1), gives
\[ 0<a^ n v_-(\omega ) =\underline{\operatorname {vol}}_{a\omega }(0) \leq \underline{\operatorname {vol}}_\theta (\rho ) \leq \underline{\operatorname {vol}}_\theta (V_\theta ). \]
In the addition step the big background is $a\omega $, and the added positive current is $(\theta -a\omega )_\rho $. Bounded mass gives finiteness of the upper volume. $\square$
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