On Camacho–sad's Theorem About the Existence of a Separatrix

Abstract

It is proved in Ann. Math. (2) 115 (1982) 579-595 that, for any germ of holomorphic nondicritic vector field in (C 2 , 0), there exists at least one separatrix (invariant analytic curve containing the origin). In Proc. Amer. Math. Soc. 125 (1997) 2649-2650 a simple criterion was given to find, at each level of the blow-up, a singular point which leads to an analytical invariant curve. In this paper we prove shortly and strictly combinatorially, the existence of a separatrix, and show that for any germ of holomorphic nondicritic vector field in (C 2 , 0), there exists at least one separatrix issuing from each connected component of the exceptional divisor of its nice blow-up with nodal corner points deleted.

Topology of singular holomorphic foliations along a compact divisor

“…, m} such that p i ∈ D cut α . The existence of a such collection follows from the below lemma whose proof is just adapted from that of the Strong Camacho-Sad Separatrix Theorem given in [14].…”

“…Indeed, it is zero if s is the attaching point of a dicritical component and it is positive if s is a nodal singularity. Then the index formulae give the inequalities j ℘ ij ≥ D i · D i and, using the terminology introduced in [14], T is a fair quasi-proper tree. We also have the inequalities ℘ ij ℘ ji ≤ 1 and T is well-balanced.…”

“…We also have the inequalities ℘ ij ℘ ji ≤ 1 and T is well-balanced. This cannot occur because of Lemma 2.1 of [14], which asserts the no existence of well balanced fair proper tree, is extended to quasi-proper trees in [14,Section 4].…”

“…Remark 1.10. The method developed in [14] immediately give a lower bound for the number of isolated separatrices for dicritical foliations in terms of the number of nodal singularities and dicritical components.…”

Topological Moduli Space for Germs of Holomorphic Foliations

“…It is enough to prove these equalities when s is an intersection point of the strict transform of a separatrix S of F with an irreducible component D of E F . Indeed according to an extension of [23] given in [16,Lemma 1.9] or in [3,Theorem 8], there is such a point on any cut-component C of E F . Thus the induction given in [15, § 7.3] will remain valid and equalities (22) will be satisfied at every singular point of the foliation.…”