Local polar invariants and the Poincaré problem in the dicritical case

Abstract

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.

Residue-type indices and holomorphic foliations

“…Off course, τ p (F) ≥ 0 and, by definition, τ p (F) = 0 if and only if SN(F) = ∅, that is, if and only if F is of second type. We introduce the following object [10,11]:…”

“…Some of these indices are calculated with respect to invariant analytic curves and we explain how to extend their definitions to formal invariant curves. We shall also present the polar excess index, introduced in [11]. In our exposition, invariant curves are identified with reduced divisors of separatrices.…”

“…Recently, second type foliations have been the object of some works. We should mention [10] -which deals with the "realization problem", that is, the existence of foliations with prescribed reduction of singularities and projective holonomy representations, [11] -which studies local polar invariants and applications to the study of the Poincaré problem for foliations -and [19] -where equisingularitiy properties are considered. Our main goal in this article is to give a characterization of second type foliations by means of residue-type indices, providing a generalization of Brunella's result.…”

“…To a germ of foliation F and a finite set of separatrices C -which can contain purely formal ones -we associate a triplet of residue-type indices: the afore mentioned CS-index and GSV-index, along with the variation index Var -that turns out to be the sum of the two fist indices (definitions in [5], [12] and [13]; see equation (14) below). We then form a quadruplet of indices by including the polar excess index ∆ ( [11] and Definition 3.1). This one is calculated by means of polar invariants and can be seen as a measure of the existence of saddle-nodes in the reduction process of F (Theorem 3.2 and Propostion 3.5).…”

Second type foliations of codimension one

“…Then ν 0 (G) ≥ ν 0 (dg) = ν 0 (Sep 0 (G)) − 1 and the equality holds if and only if G is second type [15]. This property of minimization of the algebraic multiplicity also holds in the dicritical case and a formulation for it can be seen in [10].…”