Proposition 5.6.236A2806D4E71

Proof. Local construction. Let $x\in X$, and choose holomorphic coordinates $z=(z_1,\ldots ,z_ n)$ near $F(x)$. Since $F$ is a submersion, the complex Jacobian matrix at $x$ of the $n$ components of $z\circ F$ has rank $n$ in every chart around $x$. By proposition 5.2.4(3), these components are the first $n$ coordinates of holomorphic coordinates $(z,w)\in \mathbb C^ n\times \mathbb C^{m-n}$ near $x$, in which $F$ is the projection $(z,w)\mapsto z$; shrinking, we may assume that the coordinate domain is a product $U\times W$. For a distribution $a$ on $U$, define the distribution $F^*a$ on $U\times W$ by

Integration in $w$ maps test functions on $U\times W$ continuously to test functions on $U$: the $z$-derivatives of $\int _ W\psi (\cdot ,w)\, \mathrm dV_{2m-2n}(w)$ are the integrals of the $z$-derivatives of $\psi $, by differentiation under the integral sign, and its support lies in the projection of $\operatorname{Supp}\psi $. Hence $F^*a$ is a distribution, and if $a$ is locally integrable, then $F^*a$ is the function $(z,w)\mapsto a(z)$, by Fubini's theorem. For $T=\sum T_{IJ}\, \mathrm dz_ I\wedge \mathrm d\bar z_ J$ on $U$ put $F^*T=\sum F^*T_{IJ}\, \mathrm dz_ I\wedge \mathrm d\bar z_ J$. This is a linear map from currents on $U$ to currents on $U\times W$ preserving bidegrees, continuous for weak convergence, and equal to the pullback of forms for smooth $T$. It commutes with $\partial $ and $\bar\partial $, since these act on the coefficients by differentiation, $\partial _{z_ l}F^*a=F^*\partial _{z_ l}a$, and $\partial _{w_ l}F^*a=0$ because $\int _ W\partial _{w_ l}\psi (\cdot ,w)\, \mathrm dV_{2m-2n}(w)=0$ by the fundamental theorem of calculus; similarly for the conjugate derivatives. Continuity for weak convergence holds because the coefficients of a current depend continuously on it, by their explicit description in section 5.3.

Independence and gluing. Two such local constructions near $x$ agree. Indeed, mollifying the coefficients of $T$ in a chart near $F(x)$, we obtain smooth forms $T_\varepsilon \to T$ weakly on a neighborhood of $F(x)$, as in the discussion following theorem 5.3.7. Both constructions are weakly continuous and agree with the pullback of forms on each $T_\varepsilon $, so they agree on $T$ near $x$. Consequently the local pullbacks agree on the overlaps of their domains, and by lemma 5.3.8 they glue to a unique current $F^*T$ on $X$ that agrees with each local pullback on its domain. Explicitly, if $(\chi _ c)$ is a partition of unity with compact supports subordinate to the cover of $X$ by the domains of the local pullbacks, and $(F^*T)_ c$ denotes a local pullback whose domain contains $\operatorname{Supp}\chi _ c$, then $(F^*T)(\varphi )=\sum _ c(F^*T)_ c(\chi _ c\varphi )$.

The local pullbacks are linear, preserve bidegrees and commute with $\mathrm d$, $\partial $ and $\bar\partial $; they give the pullback of forms if $T$ is smooth, and the function $a\circ F$ if $T$ is given by a locally integrable function $a$. By the uniqueness in lemma 5.3.8, $F^*T$ has the same properties. Continuity for weak convergence follows from the explicit formula, since each of the finitely many nonzero terms $(F^*T)_ c(\chi _ c\varphi )$ depends continuously on $T$.

Positivity. If $T$ is positive, the forms $T_\varepsilon $ are positive near $F(x)$, hence weakly positive at every point. By proposition 5.3.2 their pullbacks are weakly positive at every point, hence positive currents, and positivity passes to the weak limit, the local pullback of $T$ near $x$. Hence $F^*T$ is positive, by lemma 5.3.8. $\square$