Rational real algebraic models of compact differential surfaces with circle actions

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.