Azevedo Scárdua, Bruno Transversely affine and transversely projective holomorphic foliations. (English) Zbl 0889.32031 Ann. Sci. Éc. Norm. Supér. (4) 30, No. 2, 169-204 (1997). 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 \({\mathbf C} {\mathbf P}^2\). Reviewer: P.E.Newstead (Liverpool) Cited in 1 ReviewCited in 39 Documents MSC: 32S65 Singularities of holomorphic vector fields and foliations 57R30 Foliations in differential topology; geometric theory Keywords:singular holomorphic foliations; complex manifold; logarithmic foliations; Riccati foliations; transversely projective foliations × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] B. SEKE , Sur les structures transversalement affines des feuilletages de codimension (Ann. Inst. Fourier, Grenoble 30, Vol. 1, 1980 , pp. 1-29). Numdam | MR 82b:57023 | Zbl 0417.57011 · Zbl 0417.57011 · doi:10.5802/aif.773 [2] C. ANDRADE Persistência de folheações definidas por formas logarítmicas ; Thesis, IMPA, 1990 . [3] C. ANDRADE , Deformations of holomorphic foliations ; preprint : CIMAT, Guanajuato, Mexico. · Zbl 0805.32010 [4] C. CAMACHO , A. LINS NETO and P. SAD , Topological invariants and equidesingularization for holomorphic vector fields (J. of Diff. 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