Complexifications of transversely holomorphic foliations
type: Item
instance of: scholarly article
title: Complexifications of transversely holomorphic foliations
zbMATH Open document ID: 0546.57010
DOI: 10.1007/BF01455925
author: N. E. Zubov
published in: Mathematische Annalen
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/163994
review text: 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
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 32L05
Mathematics Subject Classification ID: 32Q99
zbMATH DE Number: 3870237
zbMATH Keywords: complexification of foliations
zbMATH Keywords: foliation on a principal circle bundle over a complex manifold
zbMATH Keywords: transeversely holomorphic foliation
MaRDI profile type: Publication
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OpenAlex ID: W2022601710
description: scientific article; zbMATH DE number 3870237