Luna’s slice theorem and applications. (English) Zbl 1109.14307
Wiśniewski, Jarosław A. (ed.), Algebraic group actions and quotients. Notes of the 23rd autumn school in algebraic geometry, Wykno, Poland, September 3–10, 2000. Cairo: Hindawi Publishing Corporation (ISBN 977-5945-12-7/hbk). 39-89 (2004).
This article is a study of the proof of the slice theorem of Luna for affine complex varieties. This result was originally stated and proven by D. Luna [Bull. Soc. Math. France 33, 81–105 (1973; Zbl 0286.14014)]. The slice theorem is one of the most important results in the theory of algebraic group actions. It is used to study local properties of good quotients by reductive groups.
In the present article a thorough and very readable account of this result is developed. The author starts by giving an introduction to the different notions of quotients, states and proves Zariski’s main theorem for \(G\)-morphisms, and discusses étale morphisms. Then a proof of Luna’s slice theorem is presented. Finally, three applications of the theorem are discussed. Firstly, general applications on quotients and on fixed points of reductive group actions are given. The second application concerns the local study of moduli spaces of semistable vector bundles on curves [see Y. Lazlo, Comment. Math. Helv. 71, No. 3, 373–401 (1996; Zbl 0949.14015)], and in the third application, the factoriality of local rings of a quotient and their completions is studied. It is shown that certain moduli spaces are locally factorial, but have points whose local rings have completions which are not factorial.
For the entire collection see [Zbl 1089.14002].
In the present article a thorough and very readable account of this result is developed. The author starts by giving an introduction to the different notions of quotients, states and proves Zariski’s main theorem for \(G\)-morphisms, and discusses étale morphisms. Then a proof of Luna’s slice theorem is presented. Finally, three applications of the theorem are discussed. Firstly, general applications on quotients and on fixed points of reductive group actions are given. The second application concerns the local study of moduli spaces of semistable vector bundles on curves [see Y. Lazlo, Comment. Math. Helv. 71, No. 3, 373–401 (1996; Zbl 0949.14015)], and in the third application, the factoriality of local rings of a quotient and their completions is studied. It is shown that certain moduli spaces are locally factorial, but have points whose local rings have completions which are not factorial.
For the entire collection see [Zbl 1089.14002].
Reviewer: Lucy Moser-Jauslin (MR 2006k:14082)
