PomeCorollary 13.14.19.60F41A586D0F The bounded mass property holds if and only if $\mathrm{MA}^{\mathrm{ext}}_\omega (u)(X)<+\infty $ for every $u\in \mathrm{PSH}(X,\omega )$. If the bounded mass property fails, there are potentials with infinite extended mass and trivial multiplier ideals $\mathcal I(pu)=\mathcal O_ X$, $p>0$, arbitrarily close to zero in $L^1$.
Corollary 13.14.1960F41A586D0F
Proof. The first assertion follows from proposition 13.14.16, theorem 13.14.18. For the second, take $\varepsilon U$, $0<\varepsilon \leq 1$, with $U$ as in theorem 13.14.18. It is $\omega $-plurisubharmonic and has the required exponential integrability. On $D_ t=\{ U>-t\} $, its truncation at level $\varepsilon t$ is $\varepsilon \max (U,-t)$, and
\[ \omega +\mathrm{dd}^{\mathrm{c}}\bigl (\varepsilon \max (U,-t)\bigr ) =\varepsilon \omega _{\max (U,-t)}+(1-\varepsilon )\omega . \]
Positivity of mixed products bounds the restricted measure of $\varepsilon U$ below by $\varepsilon ^ n$ times that of $U$. Its limiting mass is therefore infinite. Finally, $\| \varepsilon U\| _{L^1}\to 0$ as $\varepsilon \to 0$. $\square$
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