Variation of geometric invariant theory quotients

  • Igor V. Dolgachev
  • Yi Hu
Article

Abstract

Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parameterizing isomorphism classes of geometric objects (vector bundles, polarized varieties, etc.). The quotient depends on a choice of an ample linearized line bundle. Two choices are equivalent if they give rise to identical quotients. A priori, there are infinitely many choices since there are infinitely many isomorphism classes of linearized ample line bundles. Hence several natural questions arise. Is the set of equivalence classes, and hence the set of non-isomorphic quotients, finite? How does the quotient vary under change of the equivalence class? In this paper we give partial answers to these questions in the case of actions of reductive algebraic groups on nonsingular projective algebraic varieties. We shall show that among ample line bundles which give projective geometric quotients there are only finitely many equivalence classes. These classes span certain convex subsets (chambers) in a certain convex cone in Euclidean space, and when we cross a wall separating one chamber from another, the corresponding quotient undergoes a birational transformation which is similar to a Mori flip.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [At]
    M. Atiyah, Convexity and commuting Hamiltonians,Bull. London Math. Soc. 14 (1982), 1–15.MATHCrossRefMathSciNetGoogle Scholar
  2. [B-B]
    A. Bialynicki-Birula, Some theorems on actions of algebraic groups,Ann. of Math. 98 (1973), 480–497.CrossRefMathSciNetGoogle Scholar
  3. [B-BS]
    A. Bialynicki-Birula andA. Sommese, Quotients by C* and SL(2, C) actions,Trans. Amer. Math. Soc. 279 (1983), 773–800.MATHCrossRefMathSciNetGoogle Scholar
  4. [Bo]
    N. Bourbaki,Commutative Algebra, Berlin, New York, Springer-Verlag, 1989.MATHGoogle Scholar
  5. [Br]
    M. Brion, Sur l’image de l’application moment, in «Séminaire d’algèbre Paul Dubreil et Marie-Paule Mallivain »,Lecture Notes in Math. 1296, Paris, 1986, 177–193.MathSciNetGoogle Scholar
  6. [BP]
    M. Brion andC. Procesi, Action d’un tore dans une variété projective, in «Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory »,Progress in Mathematics 192 (1990), Birkhäuser, 509–539.MathSciNetGoogle Scholar
  7. [GM]
    M. Goresky andR. MacPherson, On the topology of algebraic torus actions, in «Algebraic Groups, Utrecht 1986 »,Lect. Notes in Math. 1271 (1986), 73–90.MathSciNetGoogle Scholar
  8. [GS]
    V. Guillemin andS. Sternberg, Birational equivalence in the symplectic category,Invent. Math. 97 (1989), 485–522.MATHCrossRefMathSciNetGoogle Scholar
  9. [He]
    W. Hesselink, Desingularization of varieties of null forms,Invent. Math. 55 (1979), 141–163.MATHCrossRefMathSciNetGoogle Scholar
  10. [Hu1]
    Y. Hu, The geometry and topology of quotient varieties of torus actions,Duke Math. Journal 68 (1992), 151–183.MATHCrossRefGoogle Scholar
  11. [Hu2]
    Y. Hu, (W, R) matroids and thin Schubert-type cells attached to algebraic torus actions,Proc. of Amer. Math. Soc. 123 No. 9 (1995), 2607–2617.MATHCrossRefGoogle Scholar
  12. [Ke]
    G. Kempf, Instability in invariant theory,Ann. of Math. 108 (1978), 299–316.CrossRefMathSciNetGoogle Scholar
  13. [KN]
    G. Kempf andL. Ness, The length of vectors in representation spaces, in «Algebraic geometry, Copenhagen 1978 »,Lecture Notes in Math. 732 (1979), Springer-Verlag, 233–243.MathSciNetGoogle Scholar
  14. [Ki1]
    F. Kirwan,Cohomology of quotients in symplectic and algebraic geometry, Princeton University Press, 1984.Google Scholar
  15. [Ki2]
    F. Kirwan, Partial desingularization of quotients of nonsinguler varieties and their Betti numbers,Annals of Math. 122 (1985), 41–85.CrossRefMathSciNetGoogle Scholar
  16. [KKV]
    F. Knop, H. Kraft, T. Vust, The Picard group of a G-variety, in «Algebraic transformation groups and invariant theory », DMV Seminar, B. 13, Birkhäuser, 1989, 77–87.Google Scholar
  17. [K1]
    S. Kleiman, Towards a numerical criterion of ampleness,Annals of Math. (2)84 (1966), 293–344.CrossRefMathSciNetGoogle Scholar
  18. [KSZ]
    M. Kapranov, B. Sturmfels, A. Zelevinsky, Quotients of toric varieties,Math. Ann. 290 (1991), 643–655.MATHCrossRefMathSciNetGoogle Scholar
  19. [Li]
    D. Lieberman, Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds, in «Séminaire François Norguet 1975/77 »,Lect. Notes in Math. 670 (1978), 140–186.MathSciNetCrossRefGoogle Scholar
  20. [MFK]
    D. Mumford, J. Fogarty, F. Kirwan,Geometric Invariant Theory, 3rd edition, Berlin, New York, Springer-Verlag, 1994.Google Scholar
  21. [Ne1]
    L. Ness, Mumford’s numerical function and stable projective hypersurfaces, in «Algebraic geometry, Copenhagen 1978 »,Lecture Notes in Math. 732 (1979), Springer-Verlag, 417–453.MathSciNetGoogle Scholar
  22. [Ne2]
    L. Ness, A stratification of the null cone via the moment map,Amer. Jour. of Math. 106 (1984), 1281–1325.MATHCrossRefMathSciNetGoogle Scholar
  23. [Re]
    M. Reid,What is a flip, preprint, Utah, 1992, 17 p.Google Scholar
  24. [Res]
    N. Ressayre,Variation de quotients en théorie des invariants, Mémoire de DEA ENS-Lyon, septembre 1996.Google Scholar
  25. [Sj]
    R. Sjamaar, Holomorphic slices, symplectic reduction and multiplicities of representations,Annals of Math. 141 (1995), 87–129.MATHCrossRefMathSciNetGoogle Scholar
  26. [Th1]
    M. Thaddeus, Stable pairs, linear systems and the Verlinde formula,Invent. Math. 117 (1994), 317–353.MATHCrossRefMathSciNetGoogle Scholar
  27. [Th2]
    M. Thaddeus, Geometric invariant theory and flips,Journal of the American Math. Society (to appear).Google Scholar

Copyright information

© Publications Mathematiques de L’I.H.E.S 1998

Authors and Affiliations

  • Igor V. Dolgachev
    • 1
  • Yi Hu
    • 2
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of MathematicsUniversity of Texas at ArlingtonArlingtonUSA

Personalised recommendations