Skip to main content
SpringerLink
Log in

EXTENSION THEOREMS FOR DIFFERENTIAL FORMS ON LOW-DIMENSIONAL GIT QUOTIENTS

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

In this paper we will show that the pull-back of any regular differential form defined on the smooth locus of a GIT quotient of dimension at most four to any resolution yields a regular differential form.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Bia lynicki-Birula, J. Świecicka, Three theorems on existence of good quotients, Math. Annalen 307 (1997), no. 1, 143–149.

    Article  MathSciNet  Google Scholar 

  2. J. F. Boutot, Singularités rationnelles et quotients par les groupes réductifs, Invent. Math. 88 (1987), no. 1, 65–68.

    Article  MathSciNet  Google Scholar 

  3. D. A. Cox, J. B. Little, H. K. Schenck, Toric Varieties, Graduate Studies in Mathematics, Vol. 124, American Mathematical Society, Providence, RI, 2011.

  4. I. Dolgachev, Lectures on Invariant Theory, London Mathematical Society Lecture Note Series, Vol. 296, Cambridge University Press, Cambridge, 2003.

  5. J. M. Drézet, Luna's slice theorem and applications, in: Algebraic Group Actions and Quotients, Hindawi Publ. Corp., Cairo, 2004, pp. 39–89.

  6. H. Esnault and E. Viehweg, Lectures on vanishing theorems, Vol. 20, Birkhäuser Verlag, Basel, 1992.

  7. B. Fantechi, L. Göttsche, Lothar, L. Illusie, S. L. Kleiman, N. Nitsure, A. Vistoli, Fundamental Algebraic Geometry. Grothendieck's FGA Explained, Mathematical Surveys and Monographs, Vol 123, American Mathematical Society, Providence, RI, 2005.

  8. P. Graf, S. J. Kovács, Potentially Du Bois spaces, J. of Singularities 8 (2014), 117–134.

    MathSciNet  MATH  Google Scholar 

  9. D. Greb, S. Kebekus, S. J. Kovács, Extension theorems for differential forms and Bogomolov–Sommese vanishing on log canonical varieties, Compositio Math. 146 (2010), no. 1, 193–219.

    Article  MathSciNet  Google Scholar 

  10. D. Greb, S. Kebekus, S. J. Kovács, T. Peternell, Differential forms on log canonical spaces, Publ. Math. Inst. de Hautes Études Sci. (2011), no. 114, 87–169.

    Article  MathSciNet  Google Scholar 

  11. R. V. Gurjar, On a conjecture of C. T. C. Wall, J. Math. Kyoto Univ.31 (1991), no. 4, 1121–1124.

    Article  MathSciNet  Google Scholar 

  12. U. Görtz, T. Wedhorn, Algebraic Geometry I, Advanced Lectures in Mathematics, Vieweg, Wiesbaden, 2010.

  13. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York, 1977.

  14. R. Hartshorne, Stable reflexive sheaves, Mathematische Annalen 254 (1980), no. 2, 121–176.

    Article  MathSciNet  Google Scholar 

  15. J. Hausen, Geometric invariant theory based on Weil divisors, Compositio Math. 140 (2004), no. 6, 1518–1536.

    Article  MathSciNet  Google Scholar 

  16. S. Iitaka, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 76, Springer-Verlag, New York, 1982.

  17. S. Kebekus, Pull-back morphisms for reflexive differential forms, Advances in Math. 245 (2013), 78–112.

    Article  MathSciNet  Google Scholar 

  18. G. R. Kempf, Some quotient varieties have rational singularities, Michigan Math. J 24 (1977), no. 3, 347–352.

    Article  MathSciNet  Google Scholar 

  19. F. C. Kirwan, Partial desingularisations of quotients of nonsingular varieties and their Betti numbers, Annals of Math. 122 (1985), no. 1, 41–85.

    Article  MathSciNet  Google Scholar 

  20. J. Koll_ar, S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, Vol. 134, Cambridge University Press, Cambridge, 1998.

  21. S. Kov_acs, Rational, log canonical, Du Bois singularities: on the conjectures of Kollár and Steenbrink, Compositio Math. 118 (1999), no. 2, 123–133.

    Article  MathSciNet  Google Scholar 

  22. S. Kov_acs, A characterization of rational singularities, Duke Math. J. 102 (2000), no. 2, 187–191.

    Article  MathSciNet  Google Scholar 

  23. H. Kraft, Geometrische Methoden in der Invariantentheorie, Friedr. Vieweg & Sohn, Braunschweig, 1984.

  24. D. Luna, R. W. Richardson, A generalization of the Chevalley restriction theorem, Duke Math. J. 46 (1979), 487–496.

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. D. Mumford, J. Fogarty, F. C. Kirwan, Geometric Invariant Theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 34, Springer-Verlag, Berlin, 1994.

  28. Y. Namikawa, Deformation theory of singular symplectic n-folds, Math. Annalen 319 (2001), no. 3, 597–623.

    Article  MathSciNet  Google Scholar 

  29. Y. Namikawa, Extension of 2-forms and symplectic varieties, J. für die Reine und Angew. Math. 539 (2001), no. 3, 123–147.

    MATH  Google Scholar 

  30. H. Pinkham, Normal surface singularities with C*action, Math. Annalen 227 (1977), no. 2, 1183–193.

  31. K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo. Sect. IA. Math. 27 (1980), no. 2, 265–291.

    MathSciNet  MATH  Google Scholar 

  32. C. S. Seshadri, Quotient spaces modulo reductive algebraic groups, Annals of Math. 95 (1972), 511–556.

    Article  MathSciNet  Google Scholar 

  33. I. R. Shafarevich, Basic Algebraic Geometry, Grundlehren der mathematischen Wissenschaften, Vol. 213, 1974, Springer-Verlag, Berlin, 1977.

    Book  Google Scholar 

  34. I. R. Shafarevich, Basic Algebraic Geometry. 2, Schemes and Complex Manifolds, Springer, Heidelberg, 2013.

    Book  Google Scholar 

  35. D. van Straten, J. H. M. Steenbrink, Extendability of holomorphic differential forms near isolated hypersurface singularities, Abh. Math. Seminar Univ. Hamburg 55 (1985), 97–110.

    Article  MathSciNet  Google Scholar 

  36. J. H. M. Steenbrink, Mixed Hodge structures associated with isolated singularities, in: Singularities, Part 2 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., Vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 513–536.

Download references

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Universität Duisburg-Essen, 45117, Essen, Germany

    S. HEUVER

Authors
  1. S. HEUVER
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. HEUVER.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

HEUVER, S. EXTENSION THEOREMS FOR DIFFERENTIAL FORMS ON LOW-DIMENSIONAL GIT QUOTIENTS. Transformation Groups 25, 81–125 (2020). https://doi.org/10.1007/s00031-019-09517-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-019-09517-8

Search

Navigation