Corollary 5.6.4FFF97DDD2CF2

Proof. (1) The local construction in the proof of proposition 5.6.2 can be carried out with the coordinates $z$ on $U$ and $(z,w)$ on $U\times W$, and it is then defined on all of $U\times W$. By the independence of the local constructions shown in that proof, $F^*T$ agrees with it near every point, hence on $U\times W$. Thus the coefficients of $F^*T$ at the basis forms containing some $\mathrm dw_ l$ or $\mathrm d\bar w_ l$ vanish, and the coefficient at $\mathrm dz_ I\wedge \mathrm d\bar z_ J$ is the distribution $\psi \mapsto \langle T_{IJ},\widehat\psi \rangle $, where $\widehat\psi (z)=\int _ W\psi (z,w)\, \mathrm dV_{2m-2n}(w)$ is a test function on $U$. Since $T_{IJ}$ is a complex Radon measure, (5.6.3) gives $\langle T_{IJ},\widehat\psi \rangle =\int _ U\widehat\psi \, \mathrm dT_{IJ}=\int \psi \, \mathrm d(T_{IJ}\otimes \lambda _ W)$ for every test function $\psi $ on $U\times W$. Hence this coefficient is the complex Radon measure $T_{IJ}\otimes \lambda _ W$.

(2) Let $x\in X$. As in the proof of proposition 5.6.2, there are holomorphic coordinates $z$ on a chart of $Y$ around $F(x)$ and holomorphic coordinates $(z,w)$ on a neighborhood $N$ of $x$, with coordinate domain a product $U\times W$, in which $F$ is the projection $(z,w)\mapsto z$. On $N$, the current $F^*T$ is the pullback of the restriction of $T$ to the open set $U$ by this projection. By (1), the coefficients of $F^*T$ in the chart $N$ are complex Radon measures. The charts $N$ obtained in this way form an atlas of $X$, so $F^*T$ has order zero, by the discussion following (5.3.6).

Now let $f$ be a locally bounded Borel function on $Y$; then $f\circ F$ is a locally bounded Borel function on $X$, since $F$ is continuous. In the coordinates above, the coefficient measures of $fT$ are the measures $fT_{IJ}$, where $f$ is written in the coordinates $z$, so by (1) the coefficient measures of $F^*(fT)$ on $N$ are the measures $(fT_{IJ})\otimes \lambda _ W$, while those of $(f\circ F)F^*T$ are the measures $(f\circ F)(T_{IJ}\otimes \lambda _ W)$; the coefficients at the basis forms containing some $\mathrm dw_ l$ or $\mathrm d\bar w_ l$ vanish for both currents. Let $g$ be a continuous function with compact support in $U\times W$. The function $g\cdot (f\circ F)$ is bounded, Borel and compactly supported, and $\widehat g(z)=\int _ Wg(z,w)\, \mathrm dV_{2m-2n}(w)$ is a bounded Borel function on $U$ with compact support. Applying (5.6.3) twice, we obtain

Hence the two currents have the same coefficient measures on $N$, and therefore agree on $N$. Since the sets $N$ cover $X$, the currents $F^*(fT)$ and $(f\circ F)F^*T$ coincide. The last assertion is the case $f=\mathbf1_ E$, since $\mathbf1_ E\circ F=\mathbf1_{F^{-1}(E)}$. $\square$