On the Topology of Foliations with a First Integral

The main objective of this article is to study the topology of the fibers of a generic rational function of the type F p Gq in the projective space of dimension two. We will prove that the action of the monodromy group on a single Lefschetz vanishing cycle δ generates the first homology group of a generic fiber of F p Gq . In particular, we will prove that for any two Lefschetz vanishing cycles δ0 and δ1 in a regular compact fiber of F p Gq , there exists a monodromy h such that h(δ0) = ±δ1. 0 Introduction Let F and G be two homogeneous polynomials in C. The following function is well-defined f = F p Gq : CP (n)\R → C f(x) = F (x) G(x)q , x = [x0; x1; · · · ; xn] where R = {F = 0} ∩ {G = 0}, deg(F ) deg(G) = q p and p and q are relatively prime numbers. We can view the fibration of f as a codimension one foliation in CP (n) given by the 1-form ω = pGdF − qFdG Let Pa denote the set of homogeneous polynomials of degree a in C . Proposition 0.1 There exists an open dense subset U of Pa × Pb such that for any (F,G) ∈ U we have: 1. {F = 0} and {G = 0} are smooth varieties in CP (n) and intersect each other transversally; 2. The restriction of f to CP (n)\({F = 0} ∪ {G = 0}) has nondegenerate critical points, namely p1, p2, . . . , pr, with distinct images in C, namely c1, c2, . . . , cr respectively. Partially supported by CNPq-Brazil