Equivariant intersection theory is similar to Borel’s equivariant cohomology. The common basic idea is simple. Let X be an algebraic scheme over a field k and let G be an algebraic group acting on X . Since invariant cycles are often too few to get a full-fledged intersection theory (e.g. to have a ring structure in smooth cases) we decide to enlarge this class to include invariant cycles not o...

Consider the Hamiltonian action of a torus on a compact twisted generalized complex manifold M. We first observe that Kirwan injectivity and surjectivity hold for ordinary equivariant cohomology in this setting. Then we prove that these two results hold for the twisted equivariant cohomology as well.

We present a functorial computation of the equivariant intersection cohomology of a hypertoric variety, and endow it with a natural ring structure. When the hyperplane arrangement associated with the hypertoric variety is unimodular, we show that this ring structure is induced by a ring structure on the equivariant intersection cohomology sheaf in the equivariant derived category. The computati...

We show that the equivariant Chow cohomology ring of a toric variety is naturally isomorphic to the ring of integral piecewise polynomial functions on the associated fan. This gives a large class of singular spaces for which localization holds in equivariant Chow cohomology with integer coefficients. We also compute the equivariant Chow cohomology of toric prevarieties and general complex hyper...

Let G be a topological group. By a G-space X we mean a topological space X together with a left action of G on X. Under the assumption that G is a compact Lie group, a discrete group or an abelian locally compact group, we constructed in [5] (see also [6]) an equivariant homology and cohomology theory, defined on the category of all G-pairs and G-maps, which both satisfy all seven equivariant E...

For a complete toric variety, we obtain an explicit formula for the localized equivariant Todd class in terms of the combinatorial data – the fan. This is based on the equivariant Riemann-Roch theorem and the computation of the equivariant cohomology and equivariant homology of toric varieties. ∗This research was supported in part by NSF grant DMS-9504522 and DMS-9803593

We give a functorial construction of a rational S 1-equivariant cohomology theory from an elliptic curve equipped with suitable coordinate data. The elliptic curve may be recovered from the cohomology theory; indeed, the value of the cohomology theory on the compactification of an S 1-representation is given by the sheaf cohomology of a suitable line bundle on the curve. The construction is eas...

The product of two Schubert cohomology classes on a Grassmannian Grk(C) has long been known to be a positive combination of other Schubert classes, and many manifestly positive formulae are now available for computing such a product (e.g., the Littlewood-Richardson rule or the more symmetric puzzle rule from A. Knutson, T. Tao, and C. Woodward [KTW]). Recently, W. Graham showed in [G], nonconst...