Normal real affine varieties with circle actions

TL;DR

This paper classifies normal affine real algebraic varieties with circle actions using a combinatorial approach based on polyhedral divisors, extending the understanding of symmetry actions in real algebraic geometry.

Contribution

It provides a complete description of such varieties with circle actions, building on and extending the polyhedral divisor framework to the real setting.

Findings

Complete classification of real affine varieties with circle actions

Extension of polyhedral divisor methods to real algebraic geometry

New insights into the structure of real varieties with circle symmetry

Abstract

We provide a complete description of normal affine algebraic varieties over the real numbers endowed with an effective action of the real circle, that is, the real form of the complex multiplicative group whose real locus consists of the unitary circle in the real plane. Our approach builds on the geometrico-combinatorial description of normal affine varieties with effective actions of split tori in terms of proper polyhedral divisors on semiprojective varieties due to Altmann and Hausen