We develop a study on local polar invariants of planar complex analytic foliations at (C 2 , 0), which leads to the characterization of second type foliations and of generalized curve foliations, as well as a description of the GSV -index. We apply it to the Poincaré problem for foliations on the complex projective plane P 2 C , establishing, in the dicritical case, conditions for the existence of a bound for the degree of an invariant algebraic curve S in terms of the degree of the foliation F . We characterize the existence of a solution for the Poincaré problem in terms of the structure of the set of local separatrices of F over the curve S. Our method, in particular, recovers the known solution for the non-dicritical case, deg(S) ≤ deg(F ) + 2.
A flag of holomorphic foliations on a complex manifold M is an object consisting of a finite number of singular holomorphic foliations on M of growing dimensions such that the tangent sheaf of a fixed foliation is a subsheaf of the tangent sheaf of any of the foliations of higher dimension. We study some basic properties of these objects and, in P n C , n ≥ 3, we establish some necessary conditions for a foliation of lower dimension to leave invariant foliations of codimension one. Finally, still in P n C , we find bounds involving the degrees of polar classes of foliations in a flag.
In this survey paper, we take the viewpoint of polar invariants to the local and global study of non-dicritical holomorphic foliations in dimension two and their invariant curves. It appears a characterization of second type foliations and generalized curve foliations as well as a description of the GSV -index in terms of polar curves. We also interpret the proofs concerning the Poincaré problem with polar invariants.
We present sufficient conditions of extending a meromorphic function which is defined outside an analytic compact curve in a complex surface. The function we deal with is a first integral for a holomorphic foliation in the whole surface. The key to extension is the study of singularities of the foliation on the complex curve.
We prove that a C ∞ equivalence between germs holomorphic foliations at (C 2 , 0) establishes a bijection between the sets of formal separatrices preserving equisingularity classes. As a consequence, if one of the foliations is of second type, so is the other and they are equisingular.2000 Mathematics Subject Classification: 32S65. 2
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.