An Oka Principle for a Parametric Infinite Transitivity Property
Abstract
It is an elementary fact that the action by holomorphic automorphisms on C^n is infinitely transitive, i.e., m-transitive for any m in N. The same holds on any Stein manifold with the holomorphic density property X. We study a parametrized case: we consider m points on X parametrized by a Stein manifold W, and seek a family of automorphisms of X, parametrized by W, putting them into a standard form which does not depend on the parameter. This general transitivity is shown to enjoy an Oka principle, to the effect that the obstruction to a holomorphic solution is of a purely topological nature. In the presence of a volume form and of a corresponding density property, similar results for volume-preserving automorphisms are obtained.
Keywords
Cite
@article{arxiv.1401.0093,
title = {An Oka Principle for a Parametric Infinite Transitivity Property},
author = {Frank Kutzschebauch and Alexandre Ramos-Peon},
journal= {arXiv preprint arXiv:1401.0093},
year = {2015}
}
Comments
Corrected minor imprecisions notably in prop 2.3, 2.4, made lemma 4.2 more precise, added interpretation of the result as a generalization of Grauert's Oka principle to principal bundles, section 5