(This is a report on joint work with S. Kaliman and M. Zaidenberg)
In this talk we give a full classification of effective $\C^*$-actions on smooth affine
surfaces up to conjugation in the full automorphism group and up
to inversion $\lambda\mapsto \lambda^{-1}$ of $\C^*$. Our main result is:
Theorem: If a smooth affine surface $V$ admits more than one $\C^*$-action up to conjugation and inversion then it belongs to one of the following classes:
(1) it is toric, (2) it is Danilov-Gizatullin, (3) it is special.
If the surface is toric then obviously the subtori of the torus acting on $V$ give many different $\C^*$-action. A Danilov-Gizatullin admits always a finite number of conjugacy classes of $\C^*$-actions. In contrast, a special surface $V$ admits a
continuous family of pairwise non-conjugated $\C^*$-actions depending on one or two parameters. We give an overview of the construction of these surfaces and sketch the proof of this result.
The isomorphism theorem for special surfaces
Lundi, 2 Février, 2009 - 11:30
Prénom de l'orateur :
Hubert
Nom de l'orateur :
FLENNER
Résumé :
Institution de l'orateur :
Ruhr University at Bochum
Thème de recherche :
Algèbre et géométries
Salle :
04