Proposition 23.2.32.FCE8FB3BF729 Let $\mathcal F$ be a foliation of a connected compact manifold $M$ with a transverse holomorphic structure of complex dimension $m\geq 1$. Let $\eta _0,\eta _1$ be transverse Kähler forms, and let $v$ be a smooth real basic function, with $\mathrm{dd}^{\mathrm{c}}v$ computed in transverse holomorphic coordinates. Suppose
\[ \eta _1=\eta _0+\mathrm{dd}^{\mathrm{c}}v, \qquad \eta _1^ m=\eta _0^ m. \]
Then $v$ is constant and $\eta _1=\eta _0$.
Proof.
The function $v$ is constant on each local plaque. On every transverse coordinate chart it satisfies
\[ \mathrm{dd}^{\mathrm{c}}v\wedge \sum _{j=0}^{m-1}\eta _1^ j\wedge \eta _0^{m-1-j}=0. \]
This is a linear elliptic equation without zeroth-order term. Let $c=\max _ Mv$. If $v=c$ at some point, the strong maximum principle on a connected transverse chart shows that $v=c$ throughout that chart, and since $v$ is basic, $v=c$ on the corresponding foliation chart. Hence $\{ v=c\} $ is nonempty, open and closed, and connectedness of $M$ proves the assertion.
$\square$
Pome
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