We describe the equivariant Chow ring of the wonderful compactification $X$ of a symmetric space of minimal rank, via restriction to the associated toric variety $Y$. Also, we show that the restrictions to $Y$ of the tangent bundle $T_X$ and its logarithmic analogue $S_X$ decompose into a direct sum of line bundles. This yields closed formulae for the equivariant Chern classes of $T_X$ and $S_X$, and, in turn, for the Chern classes of reductive groups considered by Kiritchenko.
The purpose of this article is to describe the equivariant intersection ring and equivariant Chern classes of a class of almost homogeneous varieties, namely, wonderful symmetric varieties of minimal rank; these include the wonderful compactifications of semi-simple groups of adjoint type.
The main motivation comes from questions of enumerative geometry on a spherical homogeneous space G/K. As shown by De Concini and Procesi, these questions find their proper setting in the ring of conditions C * (G/K), isomorphic to the direct limit of cohomology rings of G-equivariant compactifications X of G/K (see [DP83,DP85]). Recently, the Euler characteristic of any complete intersection of hypersurfaces in G/K has been expressed by Kiritchenko (see [Ki06]), in terms of the Chern classes of the logarithmic tangent bundle S X of any “regular” compactification X. As shown in [Ki06], these Chern classes are independent of the choice of X, and hence yield elements of C * (G/K); moreover, their determination may be reduced to the case where X is a “wonderful variety”.
In fact, it is more convenient to work with the rational equivariant cohomology ring H * G (X), from which the ordinary rational cohomology ring H * (X) is obtained by killing the action of generators of the polynomial ring H * (BG); the Chern classes of S X have natural representatives in H * G (X), the equivariant Chern classes. When X is a complete symmetric variety, the ring H * G (X) admits algebraic descriptions by work of Bifet, De Concini, Littelman, and Procesi (see [BDP90,LP90]).
Here we consider the case where X is the wonderful compactification of a symmetric space G/K of minimal rank, that is, G is semi-simple of adjoint type and rk(G/K) = rk(G)rk(K); the main examples are the groups G = (G × G)/ diag(G) and the spaces PSL(2n)/ PSp(2n). Moreover, we follow a purely algebraic approach: we work over an arbitrary algebraically closed field, and replace the equivariant cohomology
The second author thanks the IHES, the MPI and the NSA for support.
1 ring with the equivariant intersection ring A * G (X) of [EG98] (for wonderful varieties over the complex numbers, both rings are isomorphic over the rationals).
We show in Theorem 2.2.1 that the pull-back map r : A * G (X) → A * T (Y ) W K is an isomorphism over the rationals. Here T ⊂ G denotes a maximal torus containing a maximal torus T K ⊂ K with Weyl group W K , and Y denotes the closure in X of T /T K ⊂ G/K, so that Y is the toric variety associated with the Weyl chambers of the restricted root system of G/K.
We also determine the images under r of the equivariant Chern classes of the tangent bundle T X and its logarithmic analogue S X . For this, we show in Theorem 3.1.1 that the normal bundle N Y /X decomposes (as a T -linearized bundle) into a direct sum of line bundles indexed by certain roots of K; moreover, any such line bundle is the pull-back of O P 1 (1) under a certain T -equivariant morphism Y → P 1 . By Proposition 1.1.1, the product of these morphisms yields a closed immersion of the toric variety Y into a product of projective lines, indexed by the restricted roots.
In the case of regular compactifications of reductive groups, Theorem 2.2.1 is due to Littelmann and Procesi for equivariant cohomology rings (see [LP90]); it has been adapted to equivariant Chow ring in [Br98]. Here, as in the latter paper, we rely on a precise version of the localization theorem in equivariant intersection theory inspired, in turn, by a similar result in equivariant cohomology, see [GKM99]. The main ingredient is the finiteness of T -stable points and curves in X; this also plays an essential role in Tchoudjem’s description of cohomology groups of line bundles on wonderful varieties of minimal rank, see [Tc05].
For wonderful group compactifications, a more precise, “additive” description of the equivariant cohomology ring is due to Strickland, see [St06]; an analogous description of the equivariant Grothendieck group has been obtained by Uma in [Um05]. Both results may be generalized to our setting of minimal rank. However, determining generators and relations for the equivariant cohomology or Grothendieck ring is still an open question; see [Br04,Um05] for some steps in this direction.
Our determination of the equivariant Chern classes seems to be new, already in the group case; it yields a closed formula for the image under r of the equivariant Todd class of X, analogous to the well-known formula expressing the Todd class of a toric variety in terms of boundary divisors. The toric variety Y associated to Weyl chambers is considered in [Pr90,DL94], where its cohomology is described as a graded representation of the Weyl group; its realization as a general orbit closure in a product of projective lines seems to have been unnoticed.
Our results extend readily to all regular compactifications of symmetric spaces of minimal rank. Specifically, the description of the equivariant Chow ring holds un
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