Chern classes of reductive groups and an adjunction formula
[Classes de Chern des groupes réductifs et une formule d’adjonction]
Annales de l'Institut Fourier, Tome 56 (2006) no. 4, pp. 1225-1256.

Dans cet article, je construis l’analogue non compact des classes de Chern pour des fibrés vectoriel équivariants au-dessus de groupes réductifs complexes. Pour le fibré tangent, ces classes de Chern produisent une formule d’adjonction pour la caractéristique d’Euler (topologique) d’intersections complètes dans des groupes réductifs. Dans le cas d’une intersection complète qui est une courbe, cette formule donne une réponse explicite pour la caractéristique d’Euler et le genre de la courbe. Je démontre également que les classes de Chern supérieures sont nulles. La première et la dernière classe de Chern non nulle sont décrites explicitement. J’esquisse également une extension de ces résultats dans le cadre des espaces homogènes sphériques.

In this paper, I construct noncompact analogs of the Chern classes for equivariant vector bundles over complex reductive groups. For the tangent bundle, these Chern classes yield an adjunction formula for the (topological) Euler characteristic of complete intersections in reductive groups. In the case where a complete intersection is a curve, this formula gives an explicit answer for the Euler characteristic and the genus of the curve. I also prove that the higher Chern classes vanish. The first and the last nontrivial Chern classes are described explicitly. An extension of these results to the setting of spherical homogeneous spaces is outlined.

DOI : 10.5802/aif.2211
Classification : 14L30, 20G05
Keywords: Reductive groups, hyperplane section, Chern classes
Mot clés : groupes réductifs, section hyperplane, classes de Chern
Kiritchenko, Valentina 1

1 State University of New York at Stony Brook Dept. of Mathematics
@article{AIF_2006__56_4_1225_0,
     author = {Kiritchenko, Valentina},
     title = {Chern classes of reductive groups and an adjunction formula},
     journal = {Annales de l'Institut Fourier},
     pages = {1225--1256},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {56},
     number = {4},
     year = {2006},
     doi = {10.5802/aif.2211},
     zbl = {1120.14005},
     mrnumber = {2266889},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2211/}
}
TY  - JOUR
AU  - Kiritchenko, Valentina
TI  - Chern classes of reductive groups and an adjunction formula
JO  - Annales de l'Institut Fourier
PY  - 2006
SP  - 1225
EP  - 1256
VL  - 56
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2211/
DO  - 10.5802/aif.2211
LA  - en
ID  - AIF_2006__56_4_1225_0
ER  - 
%0 Journal Article
%A Kiritchenko, Valentina
%T Chern classes of reductive groups and an adjunction formula
%J Annales de l'Institut Fourier
%D 2006
%P 1225-1256
%V 56
%N 4
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2211/
%R 10.5802/aif.2211
%G en
%F AIF_2006__56_4_1225_0
Kiritchenko, Valentina. Chern classes of reductive groups and an adjunction formula. Annales de l'Institut Fourier, Tome 56 (2006) no. 4, pp. 1225-1256. doi : 10.5802/aif.2211. http://www.numdam.org/articles/10.5802/aif.2211/

[1] Bien, Frédéric; Brion, Michel Automorphisms and local rigidity of regular varieties, Compositio Math., Volume 104 (1996) no. 1, pp. 1-26 | Numdam | MR | Zbl

[2] Bifet, E.; de Concini, C.; Procesi, C. Cohomology of regular embeddings, Adv. in Math., Volume 82 (1990) no. 1, pp. 1-34 | DOI | MR | Zbl

[3] Bravi, P.; Pezzini, G. Wonderful varieties of type D (arXiv.org/math.AG/ 0410472)

[4] Brion, Michel Groupe de Picard et nombres caractéristiques des variétés sphériques, Duke Math J., Volume 58 (1989) no. 2, pp. 397-424 | DOI | MR | Zbl

[5] Brion, Michel Vers une généralisation des espaces symétriques, J. Algebra, Volume 134 (1990) no. 1, pp. 115-143 | DOI | MR | Zbl

[6] Brion, Michel The behaviour at infinity of the Bruhat decomposition, Comment. Math. Helv., Volume 73 (1998) no. 1, pp. 137-174 | DOI | MR | Zbl

[7] Brion, Michel; Kausz, Ivan Vanishing of top equivariant Chern classes of regular embeddings (preprint arxiv.org/math.AG/0503196) | MR | Zbl

[8] de Concini, C. Equivariant embeddings of homogeneous spaces, Proceedings of the International Congress of Mathematicians (Berkeley, California, USA), Volume 1,2, Amer. Math. Soc., Providence, RI (1986), pp. 369-377 | MR | Zbl

[9] de Concini, C.; Procesi, C. Complete symmetric varieties I, Invariant theory (Montecatini, 1982) (Lect. Notes in Math.), Volume 996, Springer, Berlin (1983), pp. 1-44 | MR | Zbl

[10] de Concini, C.; Procesi, C. Complete symmetric varieties II Intersection theory, Algebraic groups and related topics (Kyoto/Nagoya, 1983) (Adv. Stud. Pure Math.), Volume 6, North-Holland, Amsterdam (1985), pp. 481-513 | MR | Zbl

[11] Ehlers, F. Eine Klasse komplexer Mannigfaltigkeiten und die Auflösung einiger isolierter Singularitäten, Math. Ann., Volume 218 (1975) no. 2, pp. 127-157 | DOI | MR | Zbl

[12] Fulton, W. Intersection theory, Springer, Berlin, 1984 | MR | Zbl

[13] Gelfand, I. M.; Kapranov, M. M.; Zelevinsky, A. V. Generalized Euler integrals and A-hypergeometric functions, Adv. Math., Volume 84 (1990) no. 2, pp. 255-271 | DOI | MR | Zbl

[14] Griffiths, P.; Harris, J. Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978 | MR | Zbl

[15] Kapranov, M. Hypergeometric functions on reductive groups, Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), World Sci. Publishing, River Edge, NJ (1998), pp. 236-281 | MR | Zbl

[16] Kaveh, Kiumars Morse theory and Euler characteristic of sections of spherical varieties, Transformation Groups, Volume 9 (2004) no. 1, pp. 47-63 | DOI | MR | Zbl

[17] Kazarnovskii, B. Ya. Newton polyhedra and the Bezout formula for matrix-valued functions of finite-dimensional representations, Funct. Anal. Appl., Volume 21 (1987) no. 4, pp. 319-321 | DOI | MR | Zbl

[18] Khovanskii, A. G. Newton polyhedra, and the genus of complete intersections, Funct. Anal. Appl., Volume 12 (1978) no. 1, pp. 38-46 | MR | Zbl

[19] Kiritchenko, Valentina A Gauss-Bonnet theorem, Chern classes and an adjunction formula for reductive groups, University of Toronto, Toronto, Ontario (2004) (Ph. D. Thesis)

[20] Kleiman, S. L. The transversality of a general translate, Compositio Mathematica, Volume 28 (1974) no. 3, pp. 287-297 | Numdam | MR | Zbl

[21] Knop, Friedrich The Luna-Vust theory of spherical embeddings, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), Manoj Prakashan, Madras (1991), pp. 225-249 | MR | Zbl

[22] Knop, Friedrich Automorphisms, root systems, and compactifications of homogeneous varieties, J. Amer. Math. Soc., Volume 9 (1996) no. 1, pp. 153-174 | DOI | MR | Zbl

[23] Luna, D. Sur les plongements de Demazure, J. Algebra, Volume 258 (2002) no. 1, pp. 205-215 | DOI | MR | Zbl

[24] Richardson, R. W. Principal orbit types for algebraic transformation spaces in characteristic zero, Invent. Math., Volume 16 (1972), pp. 6-14 | DOI | MR | Zbl

[25] Rittatore, Alvaro Reductive embeddings are Cohen-Macaulay, Proc. Amer. Math. Soc., Volume 131 (2003) no. 3, pp. 675-684 | DOI | MR | Zbl

[26] Timashev, D. Equivariant compactifications of reductive groups, Sb. Math., Volume 194 (2003) no. 3–4, pp. 589-616 | DOI | MR | Zbl

Cité par Sources :