Chern classes of reductive groups and an adjunction formula

Kiritchenko, Valentina

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.

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