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.…”

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.…”

Invariant sets near singularities of holomorphic foliations

“…If z is not a saddle-node, that is, if z has eigenvalue λ = 0, then z is non-negative if Re(λ) ≥ 0 4 and it is either a node or a hyperbolic singularity. Now, it is sufficient to prove the following theorem, which improves Theorem 1.2 of [11].…”

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.…”

Invariant sets near singularities of holomorphic foliations

“…If z is not a saddle-node, that is, if z has eigenvalue λ = 0, then z is non-negative if Re(λ) ≥ 0 4 and it is either a node or a hyperbolic singularity. Now, it is sufficient to prove the following theorem, which improves Theorem 1.2 of [11].…”

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.…”

Invariant sets near singularities of holomorphic foliations

“…If z is not a saddle-node, that is, if z has eigenvalue λ = 0, then z is non-negative if Re(λ) ≥ 0 4 and it is either a node or a hyperbolic singularity. Now, it is sufficient to prove the following theorem, which improves Theorem 1.2 of [11].…”