Group completions via Hilbert schemes
Author:
Michel Brion
Journal:
J. Algebraic Geom. 12 (2003), 605-626
DOI:
https://doi.org/10.1090/S1056-3911-03-00315-1
Published electronically:
April 15, 2003
MathSciNet review:
1993758
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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.
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B M. Brion: The behaviour at infinity of the Bruhat decomposition, Comment. Math. Helv. 73 (1998), 137-174.
BP M. Brion and P. Polo: Large Schubert varieties, Representation Theory 4 (2000), 97-126.
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Ko J. Kollár: Rational curves on algebraic varieties, Ergeb. Math. 32, Springer-Verlag 1996.
L L. Lafforgue: Pavages des simplexes, schémas de graphes recollés et compactification des $\text \textrm {PGL}^{n+1}_r/\text \textrm {PGL}_r$, Invent. Math. 136 (1999), 233-271.
LP 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.
Ri R. W. Richardson: Intersections of double cosets in algebraic groups, Indag. Mathem., N. S. 3 (1992), 69-77.
Sn D. Snow: Transformation groups of compact Kähler spaces, Arch. Math. (Basel) 37 (1981), 364-371.
Sp T. A. Springer: Linear algebraic groups, Progress in Math. 9, Birkhäuser 1998.
St E. Strickland: A vanishing theorem for group compactifications, Math. Ann. 277 (1987), 165-171.
T M. Thaddeus: Complete collineations revisited, Math. Ann. 315 (1999), 489-495.
A D. Akhiezer: Lie group actions in complex analysis, Aspects of Math. E 27, Viehweg 1995.
Ba 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.
B M. Brion: The behaviour at infinity of the Bruhat decomposition, Comment. Math. Helv. 73 (1998), 137-174.
BP M. Brion and P. Polo: Large Schubert varieties, Representation Theory 4 (2000), 97-126.
De M. Demazure: Automorphismes et déformations des variétés de Borel, Invent. Math. 39 (1977), 179-186.
DP C. De Concini and C. Procesi: Complete symmetric varieties. In: Invariant Theory, 1-44, Lecture Notes in Math. 996, Springer-Verlag 1983.
DS C. De Concini and T. A. Springer: Compactification of symmetric varieties, Transform. Groups 4 (1999), 273-300.
F W. Fulton: Intersection Theory, Ergeb. der Math. 2, Springer-Verlag 1998.
K1 M. M. Kapranov: Chow quotients of Grassmannians. I, I. M. Gelfand Seminar, Adv. Soviet Math. 16 (1993), 29-110.
K2 M. M. Kapranov: Veronese curves and Grothendieck-Knudsen moduli space $\overline {M}_{0,n}$, J. Alg. Geom. 2 (1993), 239-262.
Kn F. Knop: The Luna-Vust theory of spherical embeddings, in: Proceedings of the Hyderabad Conference on Algebraic Groups, 225-250, Manoj Prakashan, Madras, 1991.
Ko J. Kollár: Rational curves on algebraic varieties, Ergeb. Math. 32, Springer-Verlag 1996.
L L. Lafforgue: Pavages des simplexes, schémas de graphes recollés et compactification des $\text \textrm {PGL}^{n+1}_r/\text \textrm {PGL}_r$, Invent. Math. 136 (1999), 233-271.
LP 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.
Ri R. W. Richardson: Intersections of double cosets in algebraic groups, Indag. Mathem., N. S. 3 (1992), 69-77.
Sn D. Snow: Transformation groups of compact Kähler spaces, Arch. Math. (Basel) 37 (1981), 364-371.
Sp T. A. Springer: Linear algebraic groups, Progress in Math. 9, Birkhäuser 1998.
St E. Strickland: A vanishing theorem for group compactifications, Math. Ann. 277 (1987), 165-171.
T 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
MR Author ID:
41725
Email:
Michel.Brion@ujf-grenoble.fr
Received by editor(s):
November 10, 2000
Published electronically:
April 15, 2003
