Theorem 10.30.428C8F709D1CC

Proof. The currents $S_ Z$ are closed, so the classes belong to the indicated affine spaces, and their compatibility follows from theorem 20.4.5, proposition 10.30.2. Take $v=\mathcal T_\theta (u;Y)$, the trace envelope of (20.3.25). By proposition 20.4.8,

on every model, which proves pseudoeffectivity in the original restricted background. The currents $S_ Z$ depend only on the ambient multiplier ideals. For a smooth real function $f$ on $Y$, the change $\theta \mapsto \theta +\mathrm{dd}^{\mathrm{c}}f$ adds $\mathrm{dd}^{\mathrm{c}}(f\circ \pi )$ to every representative, which is zero in the quotient. This proves both invariances. $\square$