Infinite families of inequivalent real circle actions on affine four-space

TL;DR

This paper constructs infinite families of inequivalent real circle actions on affine four-space, generalizing previous examples and introducing a new approach based on equivariant vector bundles.

Contribution

It develops a novel method to compare real forms of linear C*-actions, producing infinitely many non-equivalent actions on affine four-space.

Findings

Constructed infinite families of inequivalent real circle actions.

Generalized previous non-linearizable circle action example.

Introduced a new approach using equivariant vector bundles.

Abstract

The main result of this article is to construct infinite families of non-equivalent equivariant real forms of linear C*-actions on affine four-space. We consider the real form of C* whose fixed point is a circle. In [F-MJ] one example of a non-linearizable circle action was constructed. Here, this result is generalized by developing a new approach which allows us to compare different real forms. The constructions of these forms are based on the structure of equivariant O2(C)-vector bundles.