title: Finitely curved orbits of complex polynomial vector fields.
abstract: This note is about the geometry of holomorphic foliations. Let X be a polynomial vector field with isolated singularities on C2. We announce some results regarding two problems: 1. Given a finitely curved orbit L of X, under which conditions is L algebraic? 2. If X has some non-algebraic finitely curved orbit L what is the classification of X? Problem 1 is related to the following question: Let C [symbol: see text]C2 be a holomorphic curve which has finite total Gaussian curvature. Is C contained in an algebraic curve?
authors.affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, 21945-970, Brasil. albetan@im.ufrj.br
authors.forename: Albetã C
authors.initials: AC
authors.lastname: Mafra