Lemma 5.28.4.636D22C6199B Let $\pi :Z\to Y$ be a holomorphic submersion of complex manifolds. For $1\leq r\leq \dim Z$, let $u_1,\ldots ,u_ r$ be locally bounded quasi-plurisubharmonic functions on $Y$, and let $\theta _1,\ldots ,\theta _ r$ be smooth real $(1,1)$-forms on $Y$. Then
\[ \bigwedge _{i=1}^ r\bigl (\pi ^*\theta _ i+\mathrm{dd}^{\mathrm{c}}(u_ i\circ \pi )\bigr ) =\pi ^*\! \left(\bigwedge _{i=1}^ r(\theta _ i+\mathrm{dd}^{\mathrm{c}}u_ i)\right). \]
For $r>\dim Y$, the product on the left vanishes, and the right-hand side is interpreted as zero.
Proof.
Work in product coordinates in which $\pi $ is a coordinate projection. On a smaller chart of $Y$, choose smooth quadratic functions $q_ i$ such that $p_ i=u_ i+q_ i$ is plurisubharmonic, and approximate each $p_ i$ by a decreasing sequence of smooth plurisubharmonic functions. Their pullbacks have the same properties on the product chart, and for smooth forms pullback commutes with wedge products. Applying bounded decreasing continuity to the local expansions (5.28.1) on both charts gives the asserted identity, because pullback of currents by the projection is weakly continuous: pairing the pullback with a compactly supported test form amounts to pairing the original current with the integral of the test form over the fiber variables. If $r>\dim Y$, every smooth approximating product vanishes for degree reasons, and so does its limit.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.