Vector fields and foliations associated to groups of projective automorphisms

TL;DR

This paper studies Riccati foliations on complex projective bundles, providing normal forms and showing how they relate to groups of automorphisms, with a focus on their conjugacy and holonomy properties.

Contribution

It introduces normal forms for Riccati foliations on complex projective bundles and establishes a correspondence between these foliations and finitely generated automorphism groups.

Findings

Normal forms for Riccati foliations are established.

Foliations are conjugate to suspensions of automorphism groups.

Existence of Riccati foliations with prescribed holonomy groups.

Abstract

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 $G$.