Brunella, Marco On foliations of the projective space with a Kupka component. (Sur les feuilletages de l’espace projectif ayant une composante de Kupka.) (French) Zbl 1198.32015 Enseign. Math. (2) 55, No. 3-4, 227-234 (2009). If \({\mathcal F}\) is a singular holomorphic foliation of codimension 1 on \(\mathbb {CP}(n)\), with \(n\geq 3\), a connected component \(K\) of the set of singularities of \({\mathcal F}\) is said to be a Kupka component (of \({\mathcal F}\)) if, for any \(p\in K\), there is a neighbourhood \(U\) around \(p\) such that the restriction \({\mathcal F}_{|U}\) is generated by a 1-form \(\omega\) verifying \(\omega(p)=0\) and \(d\omega(p)\neq 0\). Such a component \(K\) is smooth and of codimension 2 in \(\mathbb{CP}(n)\), and, along it, \({\mathcal F}\) is locally the product \({\mathcal F}_0\times \mathbb D^{n-2}\), where \({\mathcal F}_0\) is a singular foliation in a disk \(\mathbb D^2\). The foliation \({\mathcal F}_0\) is called the transversal type of \(K\), and after X. Gómez-Mont and A. Lins-Neto [Topology 30, No. 3, 315–334 (1991; Zbl 0735.57014)] one knows that it can be represented by a linear equation \(p w dz - q z dw=0\), with \(p,q\) positive coprime integers. When \(p=q=1\), \(K\) is said to be a radial Kupka component.The foliations with a non-radial Kupka component were classified by D. Cerveau and A. Lins-Neto in [Astérisque. 222, 93–133 (1994; Zbl 0823.32014)] and O. Calvo-Andrade in [Bol. Soc. Bras. Mat., Nova Sér. 30, No. 2, 183–197 (1999; Zbl 1058.32023)]. They proved that \(K\) is a complete intersection, \(K=\{F=G=0\}\), and \({\mathcal F}\) is defined by a pencil of algebraic hypersurfaces \(pGdF - qFdG =0\) where \(p,q \in \mathbb Z^+\) and \(pq\geq 2\). Recently, O. Calvo-Andrade (unpublished) has proved that this result can be extended to the radial case: If \({\mathcal F}\) is a singular holomorphic foliation of codimension 1 on \(\mathbb{CP}(n)\), with \(n\geq 3\), with a radial Kupka component \(K\), then \(K\) is a complete intersection, \(K=\{F=G=0\}\), and \({\mathcal F}\) is defined by a pencil of algebraic hypersurfaces \(GdF -FdG =0\).The main result of the paper under review is a very interesting method to obtain a global rational first integral for a holomorphic foliation of codimension 1 on \(\mathbb{CP}(n)\), with \(n\geq 3\) from the existence of an algebraic curve that does not contain singularities of the foliation and which admits a transversal projective structure of the foliation along it (Theorem 2). As a nice application of this result, the author obtains alternative proofs of the classifications for the foliations with a Kupka component that we have mentioned above. The paper is very well written and proves interesting and nice results concerning foliations of the projective space. It is worth reading even for nonspecialists. Reviewer: Alvaro Bustinduy (Madrid) Cited in 2 ReviewsCited in 5 Documents MSC: 32S65 Singularities of holomorphic vector fields and foliations 57R30 Foliations in differential topology; geometric theory Keywords:foliations with Kupka component; projective space Citations:Zbl 0735.57014; Zbl 0823.32014; Zbl 1058.32023 × Cite Format Result Cite Review PDF Full Text: DOI