Vector fields and foliations associated to groups of projective automorphisms

2007/09/04 by Fabio H. Santos, Bruno Scardua, Santos, Fabio H. +1
Mathematics · #32M25 #32S65 #37F75 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #math.CV #math.DS #msc:32M25 #msc:32S65 #msc:37F75

paper · pdf · doi:10.48550/arxiv.0709.0546

25 pages

arxiv created 2007/09/04 · arxiv updated 2009/12/01

Abstract

We introduce and give normal forms for (one-dimensional) Riccati foliations (vector fields) on \ov \bc × \bc P(2) and \ov \bc × \ov \bcⁿ. 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 \bcⁿ, 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 × \bc P(2) for which the global holonomy is conjugate to G.

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