On Singular Holomorphic Foliations with Projective Transverse Structure

Abstract

In this paper we study holomorphic foliations with singularities having a homogeneous transverse structure of projective model (i.e., \(\mathrm{I\!P}SL(2,\mathbb {C})\) model). Our basic situation is the case of a foliation with singularities \(\mathcal F\) on a complex analytic space M of dimension two and the structure exists in the complement of some analytic subset \(S \subset M\) of codimension one. The main case occurs, as we shall see, when the analytic set is invariant by the foliation. We address both, the local and the global cases. This means two basic situations: (i) M is a projective surface (like \(M=\mathbb {C}P (2)\) or \(\overline{\mathbb {C}} \times \overline{\mathbb {C}}\)) and (ii) \(M=(\mathbb {C}^2,0)\) which means the case of germs of foliations at the origin \(0 \in \mathbb {C}^2\), having an isolated singularity at the origin. Our focus is the extension of the structure in a suitable sense. After performing a characterization of the existence of the structure in terms of suitable triples of differential forms, we consider the problem of extension of such structures to the analytic invariant set for germs of foliations and for foliations in complex projective spaces. Basic examples of this situation are given by logarithmic foliations and Riccati foliations. We also study the holonomy of such invariant sets, as a consequence of a strict link between this holonomy and the monodromy of a projective structure. These holonomy groups are proved to be solvable. Our final aim is the classification of such object under some mild conditions on the singularities they exhibit. In this work we perform this classification in the case where the singularities of the foliation are supposed to be non-dicritical and non-degenerate (more precisely, generalized curves). This case, we will see, corresponds to the transversely affine case and therefore to the class of logarithmic foliations. The more general case, which has to do with Riccati foliations, is dealt with by some extension results we prove and evoking results from Loray-Touzet-Vitorio.