Meromorphic quotients for some holomorphic G-actions

TL;DR

This paper establishes a necessary and sufficient condition for the existence of strongly quasi-proper meromorphic quotients under holomorphic actions of complex Lie groups, with applications to actions of certain subgroup products.

Contribution

It provides a new characterization for when holomorphic group actions admit meromorphic quotients and applies it to actions of product groups involving compact and complex subgroups.

Findings

Characterization of conditions for meromorphic quotients

Application to actions of groups of the form K.B

Existence results under specific subgroup conditions

Abstract

Using mainly tools from [B.13] and [B.15] we give a necessary and sufficient condition in order that a holomorphic action of a connected complex Lie group $G$ on a reduced complex space $X$ admits a strongly quasi-proper meromorphic quotient. We apply this characterization to obtain a result which assert that, when $G = K.B$ \ with $B$ a closed complex subgroup of $G$ and $K$ a real compact subgroup of $G$, the existence of a strongly quasi-proper meromorphic quotient for the $B-$action implies, assuming moreover that there exists a $G-$invariant Zariski open dense subset in $X$ which is good for the $B-$action, the existence of a strongly quasi-proper meromorphic quotient for the $G-$action on $X$.