Finitely curved orbits of complex polynomial vector fields

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 ⊂ C2 be a holomorphic curve which has finite total Gaussian curvature. Is C contained in an algebraic curve?


INTRODUCTION
Let X be a holomorphic vector fi eld with isolated singularities on C 2 and let L ⊂ C 2 be a non-singular orbit.Then L is an immersed holomorphic curve in C 2 , and its topology can be very complicated (space of ends not denumerable, infi nity genus and so on).It is also a minimal surface in R 4 to which we can associate a holomorphic Gauss map as in (Lawson 1980, Scárdua 2002).A classical theorem of Osserman states that, for a complete minimal surface in R n , the fi niteness of the total curvature is equivalent to algebraicity of its holomorphic Gauss map.See (Lawson 1980).For instance, if L is obtained from an algebraic curve C ⊂ C 2 by deleting some points then its holomorphic Gauss map is algebraic.In this work we study, for orbits of polynomial vector fi elds, the possible converses to this fact.Our main tools are the dynamics of the vector fi eld in a neighborhood of the line at infi nity L ∞ = CP 2 \ C 2 and the fact that, since L is a holomorphic curve, its corresponding holomorphic Gauss map takes values into CP 1 which can be identifi ed with L ∞ .This will relate the fi niteness of the total curvature of L with the dynamics of X close to L ∞ .
Given a polynomial vector fi eld with isolated singularities on C 2 , the dual 1-form ω = Pdy − Qdx defi nes a holomorphic foliation on C 2 whose leaves are the non-singular orbits of X and whose singular set is sing(X ).This foliation extends naturally to a one-dimensional holomorphic foliation with singularities F(X ) of the complex projective plane CP 2 and the geometry of its leaves contains important additional information on the orbits of X .Thus we shall work with the foliation F(X ) in most of our considerations.Let us recall some basic defi nitions about singularities of holomorphic foliations in dimension two.Let F a holomorphic foliation with discrete singular set sing(F) on a complex surface M. A singularity p ∈ sing(F) is called irreducible if there is an open neighborhood U of p in M where F is induced by a holomorphic differential 1-form which has one of the following types: See (Camacho and Sad 1982).An isolated singularity is called a generalized curve if its reduction process exhibits only non-degenerate singularities.It is non-dicritical if the exceptional divisor of this reduction is invariant by the foliation.See (Camacho et al. 1984).
Under suitable non-degeneracy conditions on the singularities of the foliation, a fi nitely curved orbit is algebraic as it follows from the following theorem: THEOREM 1.Let X be a polynomial vector field defined on C 2 and let L be a finitely curved orbit of X .Suppose that the singularities of F(X ) on CP 2 are non-dicritical generalized curves, then L is contained in an algebraic curve.
A Poincaré-Dulac normal form vector fi eld has non-algebraic orbits (except for the orbit contained on {y = 0}) but are fi nitely curved.This situation is described by the following theorem: THEOREM 2. Let X be a polynomial vector field on C 2 such that the singularities of F(X ) are nondicritical and in the Poincaré domain.If X has a finitely curved non-algebraic orbit then F(X ) is given by a closed rational 1-form on CP 2 .Indeed, either F(X ) is a logarithmic foliation or there is a rational map f : In particular all orbits of X have finite total curvature.

SKETCH OF THE PROOF OF THEOREMS 1 AND 2
Let us begin with a brief idea of the proof of Theorem 1.First we study the local behavior of a fi nitely curved orbit L in a neighborhood of a non-degenerated irreducible singularity p ∈ sing(X ) in C 2 .

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LEMMA 1.Let X be a holomorphic vector field on C 2 and p ∈ sing(X ) an irreducible singularity with first jet of the form If an orbit of X accumulates at p and is not contained in the union of separatrices of X through p then this orbit has infinite total curvature.
This lemma and the hypothesis in the singular set of F(X ) imply: LEMMA 2. A finitely curved orbit L of a vector field X as in Theorem 1 is contained in an analytic curve in C 2 .Indeed, we have L ⊂ L ∪ sing(X ).
The proof of Lemma 2 involves some combinatorial in the reduction of singularities for sing(F(X )) as in (Mol 2002) in order to exclude the case where L accumulates on two straight lines intersecting at the singular point on C 2 .Indeed, blowing up the singularity which is the intersection of two invariant lines we conclude that the area of the Gauss map is infi nite.The second step is to assure the analytical behavior of L in a neighborhood of L ∞ .As we have remarked above for a non-singular orbit L ⊂ C 2 the holomorphic Gauss map can be identifi ed with a map : L → L ∞ ∼ = CP 1 .Moreover, there is a leaf L ⊂ CP 2 of the foliation F(X ) such that L = L \ ( L ∩ L ∞ ).The fi niteness of the total curvature of L then implies the following: In order to fi nish the proof of Theorem 1 one applies Remmert-Stein Extension Theorem to conclude that L ⊂ CP 2 is an analytic subset of dimension one and then Chow's Theorem.SKETCH OF THE PROOF OF THEOREM 2. Let L be a nonsingular transcendental orbit of X with fi nite total curvature.We have two possibilities: CASE 1. L is closed in C 2 \ sing(F(X )).In this fi rst case we can assume that the line L ∞ is invariant by F(X ).Moreover, given a small transverse disc to L ∞ at a point q ∈ L ∞ \ sing(F(X )), L induces in an orbit which is discrete outside the origin q = ∩ L ∞ .According to Nakai's density theorem (Nakai 1994) this implies that the holonomy group of the leaf L ∞ \ sing(F(X )) is a solvable subgroup of Diff(C, 0).Now, according to the construction and classifi cation results in Sections 5 and 6 in (Scárdua 1999) this implies, taking into account the hypothesis on the singularities of F(X ), that the foliation F(X ) is given by a rational 1-form on CP 2 .According then to Section 7 in (Bracci and Scárdua 2007) this implies, always taking into account the nature of the singularities, that F(X ) is conjugated to a Poincaré-Dulac normal form as stated.CASE 2. L is not closed in C 2 and L accumulates in some invariant line E ⊂ C 2 .The same argumentation of the fi rst case can be applied to E in place of L ∞ to show that F(X ) must be conjugated to a Poincaré-Dulac normal form.