The generic rational differential equation dw/dz=P_n(z,w)/Q_n(z,w) on \mathbb{C}\mathbb{P}^2 carries no interesting transverse structure

Abstract

Let \Gamma be a non-solvable pseudogroup of holomorphic transformations in one variable fixing zero. Then for any z_0 sufficiently near zero and outside some real analytic set containing zero and depending on \Gamma, for any germ of biholomorphism \phi defined at z_0 with \phi(z_0)-z_0 sufficiently small, there exists a sequence \gamma_n \in \Gamma which tends to \phi uniformly on some neighborhood of zero. If we let \Gamma be the holonomy pseudogroup of a compact leaf, we find that holomorphic codimension-1 foliations with non-solvable holonomy admit no transverse geometric structure in addition to the conformal one. This applies, in particular, to the singular foliation induced on \mathbb{C}\mathbb{P}^2 by the differential equation dw/dz = P_n(z,w)/Q_n(z,w) where P_n and Q_n are the generic polynomials of degree n, hence the title of this paper.