Proposition 5.6.1EB9FCD4819C3

Proof. Let $\varphi $ be a test form on $Y$. By properness, the set $\operatorname{Supp}T\cap F^{-1}(\operatorname{Supp}\varphi )$ is compact. Choose a smooth function $\chi \geq 0$ with compact support on $X$ that equals one on a neighborhood of this set (proposition 5.2.6), and define

The value does not depend on $\chi $: if $\chi '$ is another choice, then $(\chi -\chi ')F^*\varphi $ is a test form whose support is disjoint from $\operatorname{Supp}T$, since $\chi =\chi '$ near $\operatorname{Supp}T\cap F^{-1}(\operatorname{Supp}\varphi )$ and $F^*\varphi =0$ outside $F^{-1}(\operatorname{Supp}\varphi )$; so $T$ vanishes on this form. For all test forms supported in a fixed compact set $L\subset Y$ a single cutoff serves, namely one equal to one near the compact set $\operatorname{Supp}T\cap F^{-1}(L)$; in particular, $F_*T$ is linear. Now let $L$ lie in a chart of $Y$, fix such a cutoff $\chi $, cover $\operatorname{Supp}\chi $ by finitely many charts of $X$ and choose a partition of unity subordinate to this cover near $\operatorname{Supp}\chi $. By the chain rule, the coefficients of the pieces of $\chi F^*\varphi $ in these charts, together with their derivatives of order at most $m'$, are bounded by a constant times $\sum _ Jp_{L,m'}(\varphi _ J)$, where the $\varphi _ J$ are the coefficients of $\varphi $ (at points mapped outside $\operatorname{Supp}\varphi $ they vanish). The estimates for $T$ in these charts therefore give the estimate in the definition of a current for $F_*T$.

Pullback preserves bidegrees, and $T$ acts on test forms of bidegree $(m-p,m-q)$, so $F_*T$ has bidegree $(n-(m-p),n-(m-q))$. For differentiation, note that $\mathrm d(\chi F^*\varphi )=\chi F^*\mathrm d\varphi +\mathrm d\chi \wedge F^*\varphi $, and $T$ vanishes on the last term because its support is disjoint from $\operatorname{Supp}T$. Since the degrees of $T$ and $F_*T$ differ by the even integer $2(m-n)$, the signs in the definitions of $\mathrm dT$ and $\mathrm dF_*T$ agree, and $\mathrm d(F_*T)=F_*(\mathrm dT)$; the same argument applies to $\partial $ and $\bar\partial $. If $\varphi $ is strongly positive, so is $\chi F^*\varphi $ by proposition 5.3.2, which proves the last assertion. $\square$