Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Group completions via Hilbert schemes

Author: Michel Brion
Journal: J. Algebraic Geom. 12 (2003), 605-626
Published electronically: April 15, 2003
MathSciNet review: 1993758
Full-text PDF

Abstract | References | Additional Information

Abstract: Let $X$ be a projective variety, homogeneous under a linear algebraic group. We show that the diagonal of $X$ belongs to a unique irreducible component ${\mathcal H}_X$ of the Hilbert scheme of $X\times X$. Moreover, ${\mathcal H}_X$ is isomorphic to the ``wonderful completion'' of the connected automorphism group of $X$; in particular, ${\mathcal H}_X$ is non-singular. We describe explicitly the degenerations of the diagonal in $X\times X$, that is, the points of ${\mathcal H}_X$; these subschemes of $X\times X$ are reduced and Cohen-Macaulay.

References [Enhancements On Off] (What's this?)

  • 1. D. Akhiezer: Lie group actions in complex analysis, Aspects of Math. E 27, Viehweg 1995.
  • 2. D. Barlet: Espace analytique réduit des cycles analytiques complexes compacts d'un espace analytique complexe de dimension finie. In: Fonctions de plusieurs variables complexes II, 1-158, Lecture Notes in Math. 482, Springer-Verlag 1975.
  • 3. M. Brion: The behaviour at infinity of the Bruhat decomposition, Comment. Math. Helv. 73 (1998), 137-174.
  • 4. M. Brion and P. Polo: Large Schubert varieties, Representation Theory 4 (2000), 97-126.
  • 5. M. Demazure: Automorphismes et déformations des variétés de Borel, Invent. Math. 39 (1977), 179-186.
  • 6. C. De Concini and C. Procesi: Complete symmetric varieties. In: Invariant Theory, 1-44, Lecture Notes in Math. 996, Springer-Verlag 1983.
  • 7. C. De Concini and T. A. Springer: Compactification of symmetric varieties, Transform. Groups 4 (1999), 273-300.
  • 8. W. Fulton: Intersection Theory, Ergeb. der Math. 2, Springer-Verlag 1998.
  • 9. M. M. Kapranov: Chow quotients of Grassmannians. I, I. M. Gelfand Seminar, Adv. Soviet Math. 16 (1993), 29-110.
  • 10. M. M. Kapranov: Veronese curves and Grothendieck-Knudsen moduli space $\overline{M}_{0,n}$, J. Alg. Geom. 2 (1993), 239-262.
  • 11. F. Knop: The Luna-Vust theory of spherical embeddings, in: Proceedings of the Hyderabad Conference on Algebraic Groups, 225-250, Manoj Prakashan, Madras, 1991.
  • 12. J. Kollár: Rational curves on algebraic varieties, Ergeb. Math. 32, Springer-Verlag 1996.
  • 13. L. Lafforgue: Pavages des simplexes, schémas de graphes recollés et compactification des $\text{\rm PGL}^{n+1}_r/\text{\rm PGL}_r$, Invent. Math. 136 (1999), 233-271.
  • 14. P. Littelmann and C. Procesi: Equivariant cohomology of wonderful compactifications, in: Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989), 219-262, Progr. Math. 92, Birkhäuser 1990.
  • 15. R. W. Richardson: Intersections of double cosets in algebraic groups, Indag. Mathem., N. S. 3 (1992), 69-77.
  • 16. D. Snow: Transformation groups of compact Kähler spaces, Arch. Math. (Basel) 37 (1981), 364-371.
  • 17. T. A. Springer: Linear algebraic groups, Progress in Math. 9, Birkhäuser 1998.
  • 18. E. Strickland: A vanishing theorem for group compactifications, Math. Ann. 277 (1987), 165-171.
  • 19. M. Thaddeus: Complete collineations revisited, Math. Ann. 315 (1999), 489-495.

Additional Information

Michel Brion
Affiliation: Université de Grenoble I, Département de Mathématiques, Institut Fourier, UMR 5582 du CNRS, 38402 Saint-Martin d’Hères Cedex, France

Received by editor(s): November 10, 2000
Published electronically: April 15, 2003

Journal of Algebraic Geometry
The Journal of Algebraic Geometry
is sponsored by the Department of Mathematical Sciences
of Tsinghua University
and is distributed by the American Mathematical Society
for University Press, Inc.
Online ISSN 1534-7486; Print ISSN 1056-3911
© 2016 University Press, Inc.
AMS Website