Proposition 5.6.56DAA1675792B

Proof. Canonical local potentials are defined on manifolds of positive dimension (theorem 5.4.3), so $Y$ has positive dimension. If $X$ has dimension zero, every $(1,1)$-current on $X$ vanishes, and $F^*T=0$. Assume that $X$ has positive dimension, let $W$ and $u$ be as in the hypothesis, and let $W_0$ be a connected component of $W$. Each point of $W_0$ has a connected neighborhood $W'\subset W_0$ in a chart of $X$ such that $F(W')$ lies in a chart of $Y$ contained in the domain of $u$. In that chart, $u$ is the canonical subharmonic representative of its class, by the remark following proposition 5.4.4, so by that proposition $u\circ F$ is either identically $-\infty $ on $W'$, or locally integrable on $W'$ with $\mathrm{dd}^{\mathrm{c}}(u\circ F)\geq 0$. No point has neighborhoods $W'$ of both kinds, because a locally integrable function is finite almost everywhere. Hence the points of $W_0$ having a neighborhood of the first kind and those having one of the second kind form two disjoint open sets covering $W_0$. Since $W_0$ is connected and $u\circ F$ is not identically $-\infty $ on $W_0$, the first set is empty. Thus $u\circ F$ is locally integrable on $W$, and $\mathrm{dd}^{\mathrm{c}}(u\circ F)\geq 0$ there.

Let $(\widetilde W,\widetilde u)$ be another pair as in the hypothesis. Near each point of the intersection of the domains of $u$ and $\widetilde u$ we have $\widetilde u=u+h$ with $h$ smooth and pluriharmonic, by theorem 5.4.3 and the remark following proposition 5.4.4. Locally $h=\operatorname{Re}g$ with $g$ holomorphic, so $h\circ F=\operatorname{Re}(g\circ F)$ is smooth and pluriharmonic, and $\widetilde u\circ F=u\circ F+h\circ F$ near each point of $W\cap \widetilde W$. Consequently $\mathrm{dd}^{\mathrm{c}}(\widetilde u\circ F)=\mathrm{dd}^{\mathrm{c}}(u\circ F)$ near each point of $W\cap \widetilde W$, hence on $W\cap \widetilde W$, since a current that vanishes near each point vanishes. By lemma 5.3.8, the local currents $\mathrm{dd}^{\mathrm{c}}(u\circ F)$, which are positive and closed because $\mathrm d\mathrm{dd}^{\mathrm{c}}=0$, glue to a unique closed positive current $F^*T$ on $X$. It is independent of the choices: the currents obtained from two systems of choices agree near each point, by what we have just shown, and hence coincide by the uniqueness in lemma 5.3.8.

If $T$ is smooth, lemma 5.4.2 gives smooth real local potentials $v$ with $T=\mathrm{dd}^{\mathrm{c}}v$; they are subharmonic, by the matrix criterion and proposition 3.1.5, hence their own canonical representatives, so we may use them as the potentials $u$. For a smooth potential $u$ the chain rule gives $\mathrm{dd}^{\mathrm{c}}(u\circ F)=F^*\mathrm{dd}^{\mathrm{c}}u=F^*T$. Now let $F$ be a submersion, and let $V$ be the domain of $u$. The restriction of $F$ to $F^{-1}(V)$ is a submersion to $V$, and its submersion pullback agrees on $F^{-1}(V)$ with that of $F$, since both are glued from the same local constructions. By proposition 5.6.2, the locally integrable function $u\circ F$ represents the submersion pullback $F^*u$, and since this pullback commutes with $\partial $ and $\bar\partial $, we obtain $\mathrm{dd}^{\mathrm{c}}(u\circ F)=\mathrm{dd}^{\mathrm{c}}F^*u=F^*\mathrm{dd}^{\mathrm{c}}u=F^*T$ on $F^{-1}(V)$, in particular on $W$. $\square$