Rational real algebraic models of compact differential surfaces with circle actions

TL;DR

This paper classifies smooth real affine algebraic surfaces with circle actions and proves that every compact differentiable surface with an $S^1$ action has a unique rational real algebraic model respecting the circle symmetry.

Contribution

It provides an algebro-geometric classification of surfaces with circle actions and establishes the uniqueness of rational models for compact surfaces with such symmetries.

Findings

Complete classification of surfaces with circle actions.

Existence of unique rational real models for compact surfaces.

Bridging differential and algebraic models via equivariant birational diffeomorphisms.

Abstract

We give an algebro-geometric classification of smooth real affine algebraic surfaces endowed with an effective action of the real algebraic circle group $\mathbb{S}^1$ up to equivariant isomorphisms. As an application, we show that every compact differentiable surface endowed with an action of the circle $S^1$ admits a unique smooth rational real quasi-projective model up to $\mathbb{S}^1$-equivariant birational diffeomorphism.