Programm des Emmy-Noether-Seminars

Fri, 23.06.2017, 14:30

Fri, 23.06.2017, 15:30
Oka Principles and the Linearization Problem

Speaker: Prof. Gerald Schwarz (Brandeis)
Host: Knop
Room: 04.363

Let G be a complex Lie group and let Q be a Stein manifold. Suppose that X and Y are holomorphic principal G-bundles over Q which admit an isomorphism $\Phi$ as topological principal G-bundles. Then the Oka principle of Grauert says that there is a homotopy $\Phi_t$ of topological isomorphisms of the principal G-bundles X and Y with $\Phi_0=\Phi$ and $\Phi_1$ biholomorphic. We prove generalizations of Grauert's Oka principle in the following situation: G is reductive, X and Y are Stein G-manifolds whose (categorical) quotients are biholomorphic to the same Stein space Q.

We give an application to the Holomorphic Linearization Problem. Let G act holomorphically on $\C^n$. When is there a biholomorphic map $\Phi$ from $\C^n$ to $\C^n$ such that the conjugate by $\Phi$ of every element of G belongs to $GL(n,\C)$? We describe a condition which is necessary and sufficient for "most" G-actions.

This is joint work with F. Kutzschebauch and F. Lárusson.