Theorem 20.4.1525F4A4E3755D

Proof. By the lemma, $N_ Z(T)=S_ Z(w;Y)$, so theorem 20.4.5 gives $g_*N_{Z'}(T)=N_ Z(T)$ for every modification $g:Z'\to Z$ over $Y$. Moreover, $g_*g^*(\pi ^*T)=\pi ^*T$. To see this, write the current as a smooth form plus $\mathrm{dd}^{\mathrm{c}}$ of a potential; the change of variables formula for the degree-one map $g$ applies both to the smooth form and to the locally integrable pulled-back potential, and pushforward commutes with $\mathrm{dd}^{\mathrm{c}}$. Therefore

Taking classes proves compatibility, and positivity of the regular parts proves pseudoeffectivity. Generic Lelong numbers are additive under addition of positive currents and scale under multiplication by nonnegative constants. Hence so are the divisorial parts and the regular parts, which proves the two identities. In the Kähler case, the regular parts and, by the $\partial \bar\partial $-lemma, their cohomology classes are exactly those of the construction cited in the statement. $\square$