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Complex-analytic quotients of algebraic \(G\)-varieties

Abstract

It is shown that any compact semistable quotient of a normal variety by a complex reductive Lie group \(G\) (in the sense of Heinzner and Snow) is a good quotient. This reduces the investigation and classification of such complex-analytic quotients to the corresponding questions in the algebraic category. As a consequence of our main result, we show that every compact space in Nemirovski’s class \(\fancyscript{Q}_G\) has a realisation as a good quotient, and that every complete algebraic variety in \(\fancyscript{Q}_G\) is unirational with finitely generated Cox ring and at worst rational singularities. In particular, every compact space in class \(\fancyscript{Q}_T\), where \(T\) is an algebraic torus, is a toric variety.

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References

  1. Alper, J.D., Easton, R.W.: Recasting results in equivariant geometry: affine cosets, observable subgroups and existence of good quotients. Transform. Groups 17(1), 1–20 (2012)

    MATH  MathSciNet  Article  Google Scholar 

  2. Artin, M.: Algebraization of formal moduli. II. Existence of modifications. Ann. Math. 2, 91 (1970)

    MathSciNet  Google Scholar 

  3. Bäker, H.: Good quotients of Mori dream spaces. Proc. Am. Math. Soc. 139(9), 3135–3139 (2011)

    MATH  Article  Google Scholar 

  4. Białynicki-Birula, A.: Quotients by actions of groups. In: Algebraic Quotients. Torus actions and cohomology. The adjoint representation and the adjoint action, Encyclopaedia of Mathematical Science, vol. 131, pp. 1–82. Springer, Berlin (2002)

  5. Białynicki-Birula, A., Sommese, A.J.: Quotients by \(\mathbb{C}^{\ast }\) and \({\rm SL}(2, \mathbb{C}) \) actions. Trans. Am. Math. Soc. 279(2), 773–800 (1983)

    MATH  Google Scholar 

  6. Białynicki-Birula, A., Sommese, A.J.: Quotients by \(\mathbb{C}^* \times \mathbb{C}^*\) actions. Trans. Am. Math. Soc. 289(2), 519–543 (1985)

  7. Białynicki-Birula, A., Świecicka, J.: On complete orbit spaces of \({\rm SL}(2)\) actions. II. Colloq. Math. 63(1), 9–20 (1992)

    MATH  MathSciNet  Google Scholar 

  8. Białynicki-Birula, A., Świecicka, J.: Open subsets of projective spaces with a good quotient by an action of a reductive group. Transform. Groups 1(3), 153–185 (1996)

    MATH  MathSciNet  Article  Google Scholar 

  9. Białynicki-Birula, A., Świecicka, J.: Three theorems on existence of good quotients. Math. Ann. 307(1), 143–149 (1997)

    MATH  MathSciNet  Article  Google Scholar 

  10. Białynicki-Birula, A., Świecicka, J.: A recipe for finding open subsets of vector spaces with a good quotient. Colloq. Math. 77(1), 97–114 (1998)

    MATH  MathSciNet  Google Scholar 

  11. Cox, D.A.: The homogeneous coordinate ring of a toric variety. J. Algebraic Geom. 4(1), 17–50 (1995)

    MATH  MathSciNet  Google Scholar 

  12. Cox, D.A., Little, J.B., Schenck, H.K.: Toric varieties. Graduate studies in mathematics, vol. 124. American Mathematical Society, Providence (2011)

    MATH  Google Scholar 

  13. Grauert, H.: Bemerkenswerte pseudokonvexe Mannigfaltigkeiten. Math. Z. 81, 377–391 (1963)

    MATH  MathSciNet  Article  Google Scholar 

  14. Grauert, H., Remmert, R.: Theory of Stein Spaces. Classics in Mathematic. Springer, Berlin (2004)

    Book  Google Scholar 

  15. Greb, D.: Compact Kähler quotients of algebraic varieties and Geometric Invariant Theory. Adv. Math. 224(2), 401–431 (2010)

    MATH  MathSciNet  Article  Google Scholar 

  16. Greb, D.: Projectivity of analytic Hilbert and Kähler quotients. Trans. Am. Math. Soc. 362, 3243–3271 (2010)

    MATH  MathSciNet  Article  Google Scholar 

  17. Greb, D.: Rational singularities and quotients by holomorphic group actions. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) X(2), 413–426 (2011)

    MathSciNet  Google Scholar 

  18. Greb, D., Heinzner, P.: Kählerian reduction in steps. In: Campbell, E., Helminck, A.G., Kraft, H., Wehlau, D. (eds.) Symmetry and Spaces—Proceedings of a workshop in honour of Gerry Schwarz, Progress in Mathematics, vol. 278, pp. 63–82. Birkhäuser, Boston (2010)

  19. Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II. Inst. Hautes Études Sci. Publ. Math. 24 (1965)

  20. Hacon, C.D., Kovács, S.J.: Classification of higher dimensional algebraic varieties. Oberwolfach seminars, vol. 41. Birkhäuser, Basel (2010)

    MATH  Book  Google Scholar 

  21. Hartshorne, R.: Ample subvarieties of algebraic varieties. Lecture Notes in Mathematics, vol. 156. Springer, Berlin (1970)

  22. Hartshorne, R.: Algebraic Geometry. Graduate texts in mathematics, vol. 52. Springer, New York (1977)

    MATH  Google Scholar 

  23. Hausen, J.: Complete orbit spaces of affine torus actions. Int. J. Math. 20(1), 123–137 (2009)

    MATH  MathSciNet  Article  Google Scholar 

  24. Hausen, J.: Three Lectures on Cox Rings. In: Torsors, Étale Homotopy and Applications to Rational Points. LMS Lecture Note Series, vol. 405, pp. 3–60. Cambridge University Press (2013)

  25. Heinzner, P.: Fixpunktmengen kompakter Gruppen in Teilgebieten Steinscher Mannigfaltigkeiten. J. Reine Angew. Math. 402, 128–137 (1989)

    MATH  MathSciNet  Google Scholar 

  26. Heinzner, P.: Geometric invariant theory on Stein spaces. Math. Ann. 289(4), 631–662 (1991)

    MATH  MathSciNet  Article  Google Scholar 

  27. Heinzner, P., Loose, F.: Reduction of complex Hamiltonian \(G\)-spaces. Geom. Funct. Anal. 4(3), 288–297 (1994)

  28. Heinzner, P., Huckleberry, A.T., Loose, F.: Kählerian extensions of the symplectic reduction. J. Reine Angew. Math. 455, 123–140 (1994)

    MATH  MathSciNet  Google Scholar 

  29. Heinzner, P., Migliorini, L., Polito, M.: Semistable quotients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26(2), 233–248 (1998)

    MATH  MathSciNet  Google Scholar 

  30. Hu, Y., Keel, S.: Mori dream spaces and GIT. Mich. Math. J. 48, 331–348 (2000)

    MATH  MathSciNet  Article  Google Scholar 

  31. Ivashkovich, S.: Limiting behavior of trajectories of complex polynomial vector fields (2010). arXiv:1004.2618

  32. King, A.D.: Moduli of representations of finite-dimensional algebras. Q. J. Math. Oxf. Ser. (2) 45(180), 515–530 (1994)

    MATH  Article  Google Scholar 

  33. Knutson, D.: Algebraic Spaces. Lecture Notes in Mathematics, vol. 203. Springer, Berlin (1971)

  34. Luna, D.: Slices étales. Bull. Soc. Math. France 33, 81–105 (1973)

  35. Luna, D.: Fonctions différentiables invariantes sous l’opération d’un groupe réductif. Ann. Inst. Fourier (Grenoble) 26(1), ix, 33–49 (1976)

  36. Lopez, A.F.: Noether-Lefschetz theory and the Picard group of projective surfaces. Mem. Am. Math. Soc. 89(438) (1991)

  37. Matsushima, Y.: Espaces homogènes de Stein des groupes de Lie complexes. Nagoya Math. J. 16, 205–218 (1960)

    MATH  MathSciNet  Google Scholar 

  38. Mumford, D.: The Red Book of Varieties and Schemes. Lecture Notes in Mathematics, vol. 1358. Springer, Berlin (1999)

  39. Mumford, D., Fogarty, J., Kirwan, F.C.: Geometric Invariant Theory. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, 2. Folge, vol. 34, 3rd edn. Springer, Berlin (1994)

  40. Nemirovski, S.: The Levi problem and semistable quotients. Complex Var. Elliptic Equ. 58(11), 1517–1525 (2013)

    MATH  MathSciNet  Article  Google Scholar 

  41. Onishchik, A.L., Vinberg, E.B.: Lie Groups and Algebraic Groups. Springer series in Soviet mathematics. Springer, Berlin (1990)

    MATH  Book  Google Scholar 

  42. Popov, V., Vinberg, E.B.: Invariant Theory. Algebraic geometry IV. In: Encyclopaedia of Mathematical Sciences, vol. 55, pp. 123–284. Springer, Berlin (1994)

  43. Rosenlicht, M.: Some basic theorems on algebraic groups. Am. J. Math. 78, 401–443 (1956)

    MATH  MathSciNet  Article  Google Scholar 

  44. Serre, J.P.: Géométrie algébrique et géométrie analytique. Ann. Inst. Fourier (Grenoble) 6, 1–42 (1955–1956)

  45. Shafarevich, I.R.: Basic Algebraic Geometry, 2nd edn. Springer, Berlin (1994)

    Book  Google Scholar 

  46. Snow, D.M.: Reductive group actions on Stein spaces. Math. Ann. 259(1), 79–97 (1982)

    MATH  MathSciNet  Article  Google Scholar 

  47. Sumihiro, H.: Equivariant completion. J. Math. Kyoto Univ. 14, 1–28 (1974)

    MATH  MathSciNet  Google Scholar 

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Acknowledgments

The author wants to thank Peter Heinzner, Christian Miebach, Stefan Nemirovski, and Karl Oeljeklaus for interesting and stimulating discussions. Furthermore, he is grateful to the organisers of the “Russian–German conference on Several Complex Variables” at Steklov Institute, during which some of these discussions took place, for the invitation and for their hospitality.

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Correspondence to Daniel Greb.

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During the preparation of this paper, the author was partially supported by the DFG-Forschergruppe 790 “Classification of algebraic surfaces and compact complex manifolds”, the DFG-Graduiertenkolleg 1821 “Cohomological Methods in Geometry”, as well as by the Baden-Württemberg-Stiftung through the “Eliteprogramm für Postdoktorandinnen und Postdoktoranden”.

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Greb, D. Complex-analytic quotients of algebraic \(G\)-varieties. Math. Ann. 363, 77–100 (2015). https://doi.org/10.1007/s00208-014-1163-y

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  • DOI: https://doi.org/10.1007/s00208-014-1163-y

Mathematics Subject Classification

  • 32M05
  • 14L24
  • 14L30