Real algebraic models of differential manifolds with circle actions

Resumen:

A classical theorem of Nash and Tognoli asserts that every compact differential manifold M has an algebraic model: a smooth real algebraic variety X (in general very far from being unique) whose real locus is diffeomorphic to M. An equivariant version of this result is known to hold for certain classes of compact manifolds endowed with a smooth action of a compact Lie group. In this talk, I will review some of the basic ideas of an algebro-combinatorial description of affine algebraic models for some manifolds with smooth actions of the circle in terms of complex graded algebras with additional Galois data.  As an illustration and a motivation for a systematic study to be done in higher dimensions, I will explain how to derive from these methods a classification of quasi-projective rational algebraic models of smooth compact surfaces with circle actions up to equivariant real birational  diffeomorphisms.

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