Proposition 13.11.14FB457F67F566

Proof. Write $T_j=\theta_j+\mathrm{dd}^{\mathrm c}\varphi_j$ by the global potential lemma, and fix a Kähler form $\omega$. Choose $C$ so large that every $\tau_j=\theta_j+C\omega$ is semipositive. The compact mass estimate proves \[\int_X\langle T_1\wedge\cdots\wedge T_p\rangle\wedge\omega^{n-p}\leq\int_X\tau_1\wedge\cdots\wedge\tau_p\wedge\omega^{n-p}<\infty.\] Thus the increasing restricted truncation measures have locally finite coefficient measures. The construction therefore defines a current everywhere; closedness of the truncation limit and the local-calculus theorem show that it is independent of the chosen potentials and patches between charts. Compactness and the Kähler form are used in the displayed uniform mass bound.