Abstract
The aim of the paper is to prove the Main Theorem which says that, in any complex normal variety with an action of a reductive groupG, there are only finitely many subsets which are maximal in the family of all openG-invariant subsets admitting a good quotient.
Similar content being viewed by others
References
[A] M. F. Atiyah,Convexity and commuting Hamiltonians, Bull. London Math. Soc.14 (1982), 1–15.
[BB, S] A. Białynicki-Birula, A. Sommese,A conjecture about quotients by tori, Adv. Stud. Pure Math.8 (1986), 59–68.
[BB, Św 1] A. Białynicki-Birula, J. Święcicka,A reduction theorem for existence of good quotients, Amer. J. Math.113 (1990) 189–201.
[BB, Św 2] A. Białynicki-Birula, J. Święcicka,Three theorems on existence of good quotients, Math. Annalen307 (1997), 143–149.
[BB, Św 3] A. Białynicki-Birula, J. Święcicka,Open subsets of projective spaces with a good quotient by an action of a reductive group, Transformation groups1, No. 3 (1996), 153–185.
[BB, Św 4] A. Białynicki-Birula, J. Święcicka,A combinatorial approach to geometric invariant theory, The Sophus Lie Memorial Conference, Oslo 1992. Proceedings. Scandinavian University Press, Oslo, 1994, 115–127.
[BB, Św 5] A. Białynicki-Birula, J. Święcicka,A recipe for finding open subsets of vector spaces with a good quotient, Coll. Math.77, No. 1 (1998), 97–113.
[Bor] A. Borel,Linear Algebraic Groups, W. A. Benjamin, New York, 1969. Russian translation: А. Борель, Линейные алuебраическиеuруппы, Мир, Москва, 1972.
[D, H] I. V. Dolgachev, Yi Hu,Variations of geometric invariant theory quotients, to appear in Publ. Mathém. de l'IHES79 (1998).
[K] J. Konarski,Decompositions of normal algebraic varieties determined by an action of a one-dimensional torus, Bull. Acad. Polon. Sci., Ser. Math., Astr., Phys.26 (1978), 285–300.
[GIT] D. Mumford, J. Fogarty,Geometric Invariant Theory, Ergeb. Math.34, Springer Verlag, 1982.
[S] C. S. Seshadri,Quotient spaces modulo reductive algebraic groups, Ann. Math.95 (1972), 511–556.
[Sum] H. Sumihiro,Equivariant completions, Journal of Math. Kyoto University14 (1974), 1–28.
[T] M. Thaddeus,Geometric invariant theory and flips, Journal of the AMS9 (1996), 691–723.
[W] J. Wŀodarczyk,Maximal quasiprojective sets and the Kleiman-Chevalley quasiprojectivity criterion, submitted to Journal of Alg. Geometry.
Author information
Authors and Affiliations
Additional information
The author is in part supported by Polish KBN Grant 2 PO3A 038 08.
Rights and permissions
About this article
Cite this article
Białynicki-Birula, A. Finiteness of the number of maximal open subsets with good quotients. Transformation Groups 3, 301–319 (1998). https://doi.org/10.1007/BF01234530
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01234530