Proper actions of Lie groups of dimension $n^2+1$ on $n$-dimensional complex manifolds

TL;DR

This paper classifies all effective, proper holomorphic actions of connected Lie groups of dimension n^2+1 on n-dimensional complex manifolds, extending previous classifications for higher and maximal group dimensions.

Contribution

It provides a complete explicit description of such actions for Lie groups of dimension n^2+1, filling a gap between known classifications for larger and maximal group dimensions.

Findings

Classified all pairs (M,G) with G of dimension n^2+1 acting effectively and properly.

Extended previous classifications for higher and maximal group dimensions.

Provided explicit descriptions of the actions and manifolds involved.

Abstract

In this paper we continue to study actions of high-dimensional Lie groups on complex manifolds. We give a complete explicit description of all pairs $(M,G)$, where $M$ is a connected complex manifold $M$ of dimension $n\ge 2$, and $G$ is a connected Lie group of dimension $n^2+1$ acting effectively and properly on $M$ by holomorphic transformations. This result complements a classification obtained earlier by the first author for $n^2+2\le\hbox{dim} G<n^2+2n$ and a classical result due to W. Kaup for the maximal group dimension $n^2+2n$.