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Transversely affine and transversely projective holomorphic foliations. (English) Zbl 0889.32031
The author considers singular holomorphic foliations \({\mathcal F}\) of codimension 1 on a complex manifold \(M\) of dimension \(n\geq 2\), having singular set \(s ({\mathcal F})\) of codimension \(\geq 2\). In particular he investigates those foliations for which the non-singular part \({\mathcal F} \mid M-s ({\mathcal F})\) is transversely affine or transversely projective, illustrating the fundamental rôle played by logarithmic foliations on \({\mathbf C} {\mathbf P}^n\) and Riccati foliations on \({\mathbf C} {\mathbf P}^2\).
The main result of Chapter I is as follows. Let \({\mathcal F}\) be a codimension 1 foliation on \({\mathbf C} {\mathbf P}^n\) which is transversely affine outside an invariant algebraic subset \(S\) of codimension 1. Suppose that \({\mathcal F}\) has reduced non-degenerate singularities in \(S\). Then \({\mathcal F}\) is a logarithmic foliation.
Chapter II concerns transversely projective foliations and includes the following central result. Let \({\mathcal F}\) be a codimension 1 foliation on \({\mathcal C} {\mathcal P}^n\) which is transversely projective outside an invariant analytic subset \(S\) of codimension 1. Then the dual foliation \({\mathcal F}^\perp\) on \({\mathbf C} {\mathbf P}^n-S\) extends to a foliation on \({\mathbf C} {\mathbf P}^n\). If \({\mathcal F}^\perp\) has a meromorphic first integral, then \({\mathcal F}\) is the rational pull-back of a Riccati foliation on \({\mathbf C} {\mathbf P}^2\).

MSC:
32S65 Singularities of holomorphic vector fields and foliations
57R30 Foliations in differential topology; geometric theory
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