Vector fields and foliations associated to groups of projective automorphisms

Details Vector fields and foliations associated to groups of projective automorphisms Vector fields and foliations associated to groups of projective automorphisms

2007-09-04

arxiv.org/abs/0709.0546v1

Mathematics

Complex Variables

25 pages

Scientific paper

We introduce and give normal forms for (one-dimensional) Riccati foliations (vector fields) on $\ov \bc \times \bc P(2)$ and $\ov \bc \times \ov \bc^n$. These are foliations are characterized by transversality with the generic fiber of the first projection and we prove they are conjugate {\em in some invariant Zariski open subset} to the suspension of a group of automorphisms of the fiber, $\bc P(2)$ or $\ov \bc^n$, this group called {\it global holonomy}. Our main result states that given a finitely generated subgroup $G$ of $\Aut(\bc P (2))$, there is a Riccati foliation on $\ov \bc \times \bc P(2)$ for which the global holonomy is conjugate to $G$.

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