×

The slice theorem for algebraic transformation groups. – Appendix: Proof of the fundamental lemma and of the slice theorem (by Friedrich Knop). (Der Scheibensatz für algebraische Transformationsgruppen. – Anhang: Beweis des Fundamentallemmas und des Scheibensatzes (von Friedrich Knop).) (German) Zbl 0722.14031

Algebraische Transformationsgruppen und Invariantentheorie, DMV Semin. 13, 89-113; Anhang: 110-113 (1989).
[For the entire collection see Zbl 0682.00008.]
The topic of this lecture is a discussion of Luna’s slice theorem [D. Luna, Bull. Soc. Math. Fr., Suppl., Mém. 33, 81-105 (1973; Zbl 0286.14014)].
The slice theorem treats the local structure of the action of a reductive algebraic group on an affine variety. The goal is to reduce this structure to get information about the action of the stabilizator on a suitable slice at the orbit of a point. The author formulates the theorem by some preliminaries and gives some applications. - The slice theorem is useful for the study of singularities and for the orbit classification.
As an appendix the reader finds a simplified version of Luna’s unpublished proof given by F. Knop.

MSC:

14L30 Group actions on varieties or schemes (quotients)
14B05 Singularities in algebraic geometry
57S25 Groups acting on specific manifolds
14M17 Homogeneous spaces and generalizations