Abstract
Let π : X → Y be a good quotient of a smooth variety X by a reductive algebraic group G and 1≤k≤ dim (Y) an integer. We prove that if, locally, any invariant horizontal differential k-form on X (resp. any regular differential k-form on Y) is a Kähler differential form on Y then codim(Y sing)>k+1. We also prove that the dualizing sheaf on Y is the sheaf of invariant horizontal dim(Y)-forms.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Boutot, J.-F.: Singularités rationnelles et quotients par les groupes réductifs, Invent. Math. 88(1) (1987), 65–68.
Brion, M.: Differential forms on quotients by reductive group actions, Proc. Amer. Math. Soc. 126(9) (1998), 2535–2539.
Cox, D. A.: The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4(1) (1995), 17–50.
El Zein, F.: Complexe dualisant et applications a' la classe fondamentale d'un cycle, Bull. Soc. Math. France Mém. 58 (1978), 93.
Encinas, S. and Villamayor, O.: Good points and constructive resolution of singularities, Acta Math. 1816(1) (1998), 109–158.
Flenner, H.: Extendability of differential forms on nonisolated singularities, Invent. Math. 946(2) (1988), 317–326.
Fogarty, J.: Invariant differentials, In: Algebraic Geometry and Commutative Algebra, Vol. I, Kinokuniya, Tokyo, 1988, pp. 65–72.
Grothendieck, A.: 1967, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 361.
Hartshorne, R.: Residues and Duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Math. 20, Springer-Verlag, Berlin.
Jamet, G.: Obstruction au prolongement des formes différentielles régulie' res et codimension du lieu singulier, PhD thesis, UniversitéParis VI. http:// www.math.jussieu.fr/_jamet/Pub/Thesis/.
Kersken, M.: Cousinkomplex und Nennersysteme, Math. Z. 1826(3) (1983), 389–402.
Kersken, M.: Der Residuenkomplex in der lokalen algebraischen und analytischen Geometrie, Math. Ann. 265(4) (1983), 423–455.
Kersken, M.: Reguläre Differentialformen, Manuscripta Math. 46(1-3) (1984), 1–25.
Knop, F.: Der kanonische Modul eines Invariantenrings, J. Algebra 1276(1) (1989), 40–54.
Kunz, E. and Waldi, R.: Regular Differential Forms, Amer. Math. Soc., Providence, RI, 1988.
Luna, D.: Slices étales, Bull. Soc. Math. France, Paris, Mém. 33 (1973), 81–105.
Mumford, D. and Fogarty, J.: Geometric Invariant Theory, 2nd edn, Springer-Verlag, Berlin, 1982.
Serre, J.-P.: Groupes finis d'automorphismes d'anneaux locaux réguliers, In: Colloque d'algèbre (Paris, 1967), Exp. 8. Secrétariat mathématique, Paris, 1968, p. 11.
Shephard, G. C. and Todd, J. A.: Finite unitary reflection groups, Canad. J. Math. 6, (1954), 274–304.
van Straten, D. and Steenbrink, J.: Extendability of holomorphic differential forms near isolated hypersurface singularities, Abh. Math. Sem. Univ. Hamburg 55 (1985), 97–110.
Vetter, U.: Öußere Potenzen von Differentialmoduln reduzierter vollständiger Durchschnitte, Manuscripta Math. 2 (1970), 67–75.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Jamet, G. Differential Forms and Smoothness of Quotients by Reductive Groups. Compositio Mathematica 133, 151–171 (2002). https://doi.org/10.1023/A:1019630928321
Issue Date:
DOI: https://doi.org/10.1023/A:1019630928321