Corollary 10.30.59157A5A52988

Proof. Let $\pi :Z\to Y$ be the structure map and put $\pi '=\pi \circ g$. Abbreviate $S_ Z=S_ Z(u;Y)$, and use lemma 5.38.4 to write $S_ Z=B_ Z+\mathrm{dd}^{\mathrm{c}}v_ Z$ with $B_ Z$ smooth and closed and $v_ Z$ quasi-plurisubharmonic. The affine class $\mathbb B_ Z$ has the smooth representative $\pi ^*\theta -B_ Z$. Its pullback is therefore the class of $\pi '^*\theta -g^*S_ Z$, since the two representatives differ by $\mathrm{dd}^{\mathrm{c}}(v_ Z\circ g)$. Substituting $S_{Z'}=g^*S_ Z+[E_ g]$ gives the asserted identity. Both representatives have exterior derivative $\pi '^*\mathrm{d}\theta $, because the subtracted currents are closed, so the subtraction takes place within $\mathscr A_{Z'}(\theta )$. $\square$