An Oka Principle for a Parametric Infinite Transitivity Property

Abstract

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 ∈ 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.

The first thirty years of Andersen-Lempert theory

“…The following parametric version of Proposition 3.1, due to Ramos-Peon and Ugolini [151], generalises earlier results of Kutzschebauch and Ramos-Peon [129] and Ugolini [171]. It concerns interpolation of jets by holomorphic automorphisms at finitely many points of a Stein manifold X with the density property, where the jets and the interpolating automorphisms of X depend holomorphically on a parameter in another Stein manifold W .…”

“…The simplest case is a trivial fibration W ×X → W, (w, x) → w, where X is a Stein manifold with the density property and the parameter space W is a Stein manifold. For this case the fibred density property is easy to prove [123,129]. The implication for the corresponding group of holomorphic automorphisms (in a simple situation of a projection…”

Parametric jet interpolation for Stein manifolds with the density property

“…We point out that we just need to assume for the parametrized family of jets to be null-homotopic and not homotopic to the jet of the identity as in (i), this is equivalent as Y is path-connected when X is. This is the strategy followed in [KRP17] for the pointwise interpolation case (i.e. k = 0).…”

A parametric jet-interpolation theorem for holomorphic automorphisms of C^n

“…After proving Proposition 2.5 and Theorem 1.1, we will present a corollary of our main result and of the recent result of Kutzschebauch and Ramos-Peon [13] concerning the possibility of interpolating a holomorphically moving family of points by a holomorphic family of automorphisms. Combining their result with our Theorem 1.1 for a finite number of points yields a theorem on interpolating a holomorphically moving collection of points, as well as finite order jets at these points, by a holomorphic family of automorphisms; see Corollary 2.8.…”