Construction of vector fields and Riccati foliations associated to groups of projective automorphisms

Abstract

Our main result states that given a finitely generated subgroup G G of Aut ⁡ ( C P ( 2 ) ) \operatorname {Aut}(\mathbb {C} P (2)) , there is an algebraic foliation F \mathcal {F} on a complex projective 3 3 -manifold M 3 M^3 with a bundle structure over C P ( 1 ) \mathbb {C} P(1) and fiber C P ( 2 ) \mathbb {C} P(2) , such that F \mathcal {F} is transverse to almost every fiber of the bundle and with global holonomy conjugate to G G .