Analytic Hilbert Quotients

Summary

We give a systematic treatment of the quotient theory for a holomorphic action of a reductive group G = K^ℂ on a not necessarily compact Kählerian space X. This is carried out via the complex geometry of Hamiltonian actions and in particular uses strong exhaustion properties of K-invariant plurisubharmonic potential functions.

The open subset X(μ) of momentum semistable points is covered by analytic Luna slice neighborhoods which are constructed along the Kempf- Ness set μ-1﹛0﹜. The analytic Hilbert quotient X (μ) → X(μ)//G is defined on these Stein neighborhoods by complex analytic invariant theory. If X is projective algebraic, then these quotients are those given by geometric invariant theory. The main results here appear in various contexts in the literature. How- ever, a number of proofs are new and we hope that the systematic treatment will provide the nonspecialist with basic background information as well as details of recent developments.

1. Introduction

As the title indicates, we focus here on a certain quotient construction for group actions on complex spaces. Our attention is primarily devoted to actions of (linear) reductive complex Lie groups, i.e., complex matrix groups which are complexifications G = K^ℂ of their maximal compact subgroups.