PomeLemma 11.22.4.3CD7629C6061 Let $Y$ be a complex manifold of dimension $\ell \geq 1$, let $\eta $ be a Hermitian form on $Y$, and let $\pi :Z\to Y$ be a holomorphic fiber bundle whose fiber is either $\mathbb C$ or a compact connected Riemann surface. If $S$ is a positive real $(1,1)$-current on $Z$ with $\mathrm{dd}^{\mathrm{c}}S=0$ and $S\wedge \pi ^*\eta ^\ell =0$, then there is a unique positive $\mathrm{dd}^{\mathrm{c}}$-closed $(1,1)$-current $R$ on $Y$ such that $S=\pi ^*R$. If $S$ is closed, then $R$ is closed.
Lemma 11.22.43CD7629C6061
Proof. Fix a local trivialization $\pi ^{-1}(U)=U\times F$ of the bundle, with base coordinates $w=(w_1,\ldots ,w_\ell )$ and a local fiber coordinate $z$. The coefficient measures of $S$ have a positive Hermitian matrix of densities relative to their trace measure. The identity $S\wedge \pi ^*\eta ^\ell =0$ makes the vertical diagonal entry zero, and the inequality $|s_{\alpha \bar z}|^2\leq s_{\alpha \bar\alpha }s_{z\bar z}$ then removes all mixed entries. Thus
\[ S=\frac{\mathrm{i}}{2\pi }\sum _{\alpha ,\beta =1}^\ell s_{\alpha \bar\beta }(w,z)\, \mathrm{d}w_\alpha \wedge \mathrm{d}\bar w_\beta , \]
with distribution coefficients. In the equation $\partial \bar\partial S=0$, the coefficient containing both vertical differentials and the horizontal pair $(\alpha ,\beta )$ gives $\partial _ z\partial _{\bar z}s_{\alpha \bar\beta }=0$.
Fix $v\in \mathbb C^\ell $ and a nonnegative smooth function $\xi $ with compact support in $U$. Pairing the positive coefficient distribution $\sum _{\alpha ,\beta }s_{\alpha \bar\beta }v_\alpha \bar v_\beta $, in the base variables, with $\xi $ times the Euclidean volume form defines a nonnegative harmonic distribution on the whole fiber $F$. Its definition does not depend on the fiber coordinate, since the remaining differentials are horizontal. Weyl's lemma makes it a nonnegative harmonic function. Such a function is constant on $\mathbb C$ by the Liouville theorem for harmonic functions, and on a compact connected curve by the maximum principle.
Polarization in $v$ and linearity in the base test function show that every coefficient $s_{\alpha \bar\beta }$ is the pullback of a distribution $r_{\alpha \bar\beta }$ on $U$. To identify this distribution, pair against any nonnegative smooth fiber density of total integral one; this also proves positivity of the resulting current $R_ U$ on $U$ and injectivity of the pullback. Consequently $\mathrm{dd}^{\mathrm{c}}S=\pi ^*(\mathrm{dd}^{\mathrm{c}}R_ U)=0$ implies $\mathrm{dd}^{\mathrm{c}}R_ U=0$. On overlaps, injectivity makes the currents $R_ U$ agree, so they glue to a current $R$, and injectivity also proves uniqueness. If $\mathrm{d}S=0$, the same argument gives $\mathrm{d}R=0$. $\square$
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