##### Department of Mathematics,

University of California San Diego

****************************

### Math 295 - Mathematics Colloquium

## Gerald Schwarz

#### Brandeis University

## Oka Principles and the Linearization Problem.

##### Abstract:

This is a talk for a general audience. Let $G$ be a complex Lie group and let $Q$ be a Stein manifold (closed complex submanifold of some $\Bbb C^n$). Suppose that $X$ and $Y$ are holomorphic principal $G$-bundles over $Q$ which admit an isomorphism $\Phi$ as topological principal $G$-bundles. Then the famous 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 $\Bbb C^n$. When is there a biholomorphic map $\Phi\colon \Bbb C^n \to \Bbb C^n$ such that $\Phi^{-1} \circ g \circ \Phi \in \rm{GL}(n,C)$ for all $g \in G$? We describe a condition which is necessary and sufficient for ``most" $G$-actions. This is joint work with F. Kutzschebauch and F. Larusson.

Host: Alvaro Pelayo

### April 7, 2016

### 4:00 PM

### AP&M 6402

****************************