Complexifications of transversely holomorphic foliations
Let \({\mathcal F}\) be a transeversely holomorphic foliation on a paracompact manifold X, of dimension p and complex codimension n. This means that \({\mathcal F}\) is given by an open covering \(\{U_ i\}_{i\in I}\) and local submersions \(f_ i: U_ i\to {\mathbb{C}}^ n\) with fibers of dimension p such that, for i,\(j\in I\), there is a holomorphic isomorphism \(g_{ji}\) of open sets of \({\mathbb{C}}^ n\) such that \(f_ j=g_{ji}\cdot f_ i\) on \(U_ i\cap U_ j\). - A complexification of \({\mathcal F}\) is a complex analytic manifold \(\hat X\) of complex dimension \(n+p\) with a holomorphic foliation \(\hat {\mathcal F}\) of codimension n, and an embedding j: \(X\to\hat X\) such that \({\mathcal F}=j^{-1}(\hat F)\) as transversely holomorphic foliations, and such that the images of the leaves of \({\mathcal F}\) by j are totally real in the leaves of \(\hat {\mathcal F}.\) We show that a complexification of \({\mathcal F}\) always exists when the codimension is one, but that in general complexifications do not exist when the codimension is greater than one. We also show that the natural transversely holomorphic foliation on a principal circle bundle X over a complex manifold M admits a complexification if and only if the associated line bundle over M comes from a holomorphic line bundle
foliation on a principal circle bundle over a complex manifold
scientific article; zbMATH DE number 3870237