Invariant meromorphic functions on Stein spaces

TL;DR

This paper develops tools to study invariant meromorphic functions on Stein spaces with reductive group actions, constructing quotients and analyzing orbit separation, with applications to almost homogeneous spaces and invariants.

Contribution

It constructs Rosenlicht-type quotients for Stein spaces under reductive group actions and explores the relation between holomorphic and meromorphic invariants.

Findings

Invariant meromorphic functions separate orbits in general position.

Constructed quotients generalize classical Rosenlicht quotients.

Established a weak equivariant Narasimhan embedding theorem.

Abstract

In this paper we develop fundamental tools and methods to study meromorphic functions in an equivariant setup. As our main result we construct quotients of Rosenlicht-type for Stein spaces acted upon holomorphically by complex-reductive Lie groups and their algebraic subgroups. In particular, we show that in this setup invariant meromorphic functions separate orbits in general position. Applications to almost homogeneous spaces and principal orbit types are given. Furthermore, we use the main result to investigate the relation between holomorphic and meromorphic invariants for reductive group actions. As one important step in our proof we obtain a weak equivariant analogue of Narasimhan's embedding theorem for Stein spaces.