Stability of the transversal Kählerian character (Q1380514)

The main result in this paper is the following: {Theorem}. Let \({\mathcal F}\) be a Hermitian homologically oriented foliation on a compact manifold \(M\) and \({\mathcal F}_t\) a deformation of \({\mathcal F}\) of fixed differential type parametrized on the neighborhood \(T\) of \({\mathcal O}\) in \(R^d\) by the transversally holomorphic foliation. If the Hermitian transversal metric \(\sigma \) of \({\mathcal F}= {\mathcal F}_0\) is Kählerian, then there exists \(\varepsilon >0\) such that for all \(t\in T\), \(|t|<\varepsilon \), the foliation \({\mathcal F}_t\) posseses a transversal Kählerian metric \(\sigma _t\), such that \(\sigma _0=\sigma \), which depends differentiably on \(t\) for \(|t|< \varepsilon \). The authors give an example for the existence of a transversally Kählerian foliation which allows nontrivial deformations of fixed differential type.