Local polar invariants for plane singular foliations

Abstract

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.

Differentiable equisingularity of holomorphic foliations

“…We say that a saddle-node singularity for F is tangent if its weak separatrix is contained in E. Non-tangent saddle-nodes are also known as well-oriented. The following definition is due to J.-F. Mattei and E. Salem (see [6] and also [2], [3] [4]): Definition 3.1. A foliation F at (C 2 , 0) is of second type if there are no tangent saddlenodes in its reduction of singularities.…”

Residue-type indices and holomorphic foliations

“…We say that B is invariant by η if γ * η ≡ 0. In this case, we define the polar intersection number of η and B at p (see [6,11]) as the generic value of (P η , B) p = (P η (a,b) , B) p = ord t=0 ((aP + bQ) • γ) for (a : b) ∈ P 1 . This is an ingredient for the following definition: Definition 3.1.…”

“…Let B be a branch of separatrix and F be a reduced balanced equation of separatrices adapted to B. The polar excess index [6,11] of F with respect to B is the integer…”

“…Suppose that an F-invariant algebraic curve S ⊂ P 2 is defined by a homogeneous polynomial equation P = P 1 P 2 • • • P n = 0, where each polynomial P i is irreducible of degree d i . Suppose further that that F is non-dicritical at each point q ∈ S. Then the following statements are equivalent [3,6,8]:…”

Local polar invariants and the Poincaré problem in the dicritical case

“…In a recent work [7], F. Cano, N. Corral and the second author developed a study of local polar invariants obtaining, in the non-dicritical case, a characterization of generalized curves and of foliations of second type as well as an expression of the GSV -index in terms of these invariants. Essentially, the technique therein consists on calculating the intersection number between a generic polar curve of a local foliation F and a curve of formal separatrices C. The same number is produced for the formal "reference foliation" having the local equation of C as a first integral.…”