Construction of invariant curves for singular holomorphic vector fields

Camacho and Sad proved the existence of invariant analytic curves for germs of singular holomorphic foliations F over a two dimensional complex analytic variety M . Their proof is only of existential nature. Here we provide a simple constructive proof by giving criteria to choose a singular point at each blowing-up that follows an analytic invariant curve. Our algorithm is founded on the stability by blowing-up of the property (?) introduced in the following definition. Definition. Consider a singular holomorphic foliation F over a two dimensional complex analytic variety M , a normal crossings divisor E over M and a point q ∈ E. We say that the triple (F , E, q) has the property (?) if and only if one of the following properties holds: (?)-1: The point q lies exactly in one irreducible component S of E, which is invariant for F and the index iq(F , S) / ∈ Q(≥0) = {r ∈ Q; r ≥ 0}. (?)-2: The point q lies in two irreducible components S+ and S− of E (call this point a “corner”), both are invariant curves and there is a real number a > 0 such that: iq(F , S+) ∈ Q(≤−a) = {r ∈ Q; r ≤ −a}, iq(F , S−) / ∈ Q(≥−1/a) = {r ∈ Q; r ≥ −1/a}. (?)-3: The point q lies exactly in one irreducible component S of E, it is a nonsingular point of F and S is transversal to F at q. (The definition and basic properties of the index can be found in [1]). Remark. If we have the property (?)-2, then q is not a simple (irreducible) singularity. If we have either (?)-1 and q is a simple singularity or (?)-3, then there is a nonsingular analytic invariant curve Γ through q transversal to E. Theorem. Assume that (F , E, q) satisfies either (?)-1 or (?)-2. Consider the blowing-up π : M ′ → M at the point q. Let F ′ be the strict transform of F by π. Put D = π−1(q) and E′ = π−1(E). Then there is a point q′ ∈ D such that (F ′, E′, q′) satisfies the property (?). Received by the editors October 24, 1995 and, in revised form, March 19, 1996. 1991 Mathematics Subject Classification. Primary 34A05; Secondary 32S65. This work was supported by the I.A.S. under NSF grant # DMS-9304580. c ©1997 American Mathematical Society 2649 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use