Proposition 10.17.7719EA7B7039B

Proof. Positive trace mass means $\int _ Zb_ v^ k>0$. Since $b_ v$ is nef, (10.16.7) gives a current $K=\eta _ v+\mathrm{dd}^{\mathrm{c}}a\geq \delta \kappa $ in $b_ v$, where $\kappa $ is a Kähler form and $\delta >0$. The current

is then a Kähler current. Since $a$ is bounded above on $Z$ and $w$ is bounded below, its potential $p=\ell +a$ satisfies $p\leq \ell +w+A=v_\pi +A$ for a constant $A$. If $b_ v$ has a smooth positive representative, $a$ can be chosen smooth, and then $p-v_\pi =a-w$ is bounded.

For smooth $Y$, take $\alpha _ Y=\alpha _0|_ Y$ when $v$ is a restriction. By proposition 5.43.18, $p=q\circ \pi $ for some $q\in \mathrm{PSH}(Y,\alpha _ Y)$. The current $\alpha _ Y+\mathrm{dd}^{\mathrm{c}}q$ is strict by (10.16.8), and since $\pi $ is surjective, each inequality between potentials descends. The hypothesis of a smooth positive representative refers to the specified resolution: the pullback of a Kähler class under a further blowup need not be Kähler. $\square$