The theory of characteristic classes of vector bundles and smooth manifolds plays an important role in the theory of smooth manifolds. An investigation of reasonable notions of characteristic classes of singular spaces started with a systematic study of singular spaces such as singular algebraic varieties. We give a quick survey of characteristic classes of singular varieties, mainly focusing on the functorial aspects of some important ones such as the singular versions of the Chern class, the Todd class and Thom–Hirzebruch’s L-class. Further we explain our recent “motivic” characteristic classes, which in a sense unify these three different theories of characteristic classes. We also discuss bivariant versions of them and characteristic classes of proalgebraic varieties, which are related to the motivic measures/integrations. Finally we explain some recent work on “stringy” versions of these theories, together with some references for “equivariant” counterparts.

Abstract. The equivariant and ordinary cohomology rings of Hilbert schemes of points on the minimal resolution C 2 //Γ for cyclic Γ are studied using vertex operator technique, and connections between these rings and the class algebras of wreath products are explicitly established. We further show that certain generating functions of equivariant intersection numbers on the Hilbert schemes and related moduli spaces of sheaves on C 2 //Γ are τ-functions of 2-Toda hierarchies. 1.

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

invariants of genus zero on a Grassmann variety are equal to triple intersections computed in the (equivariant) K-theory of a two-step flag manifold, thus generalizing an earlier result of Buch, Kresch, and Tamvakis. In the process we show that the Gromov-Witten variety of curves passing through 3 general points is irreducible and rational. Our applications include Pieri and Giambelli formulas for the quantum K-theory ring of a Grassmannian, which determine the multiplication in this ring. Our formula for Gromov-Witten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of Chaput, Manivel, and Perrin. 1.

Abstract. In this paper we consider the K-theory of smooth algebraic stacks, establish λ and Adams operations and show that the higher K-theory of such stacks is always a pre-λ-ring and is a λ-ring if every coherent sheaf is the quotient of a vector bundle. As a consequence we are able to define Adams operations and absolute cohomology for smooth algebraic stacks satisfying this hypothesis. We also define a Riemann-Roch transformation and prove a Riemann-Roch theorem for strongly projective morphisms between smooth stacks. When the stack is a scheme, all these are shown to reduce to the corresponding results for schemes. Table of Contents 1.

Abstract. In this paper we give an explicit formula for the Riemann-Roch map for singular schemes which are quotients of smooth schemes by diagonalizable groups. As an application we obtain a simple proof of a formula for the Todd class of a simplicial toric variety. An equivariant version of this formula was previously obtained for complete simplicial toric varieties by Brion and Vergne [BV] using different techniques. 1.

Abstract. The goal of this paper is to prove the equivariant version of Bloch’s Riemann-Roch isomorphism between the higher algebraic K-theory and the higher Chow groups of smooth varieties. We show that for a linear algebraic group G acting on a smooth variety X, although there is no Chern character map from the equivariant K-groups to equivariant higher Chow groups, there is indeed such a map KG i (X)⊗R(G) ̂ R(G) ch − → CH ∗ G (X, i)⊗S(G) ̂ S(G) with rational coefficients, which is an isomorphism. This implies the Riemann-Roch isomorphism ̂ KG G bτ X i (X) −− → ̂ CH ∗ G (X, i). The case i = 0 provides a stronger form of the Riemann-Roch theorem of Edidin and Graham (cf. [6]). 1.

Abstract. We provide a formula describing the G-module structure of the Hurwitz-Hodge bundle for admissible G-covers in terms of the Hodge bundle of the base curve, and more generally, for describing the G-module structure of the push-forward to the base of any sheaf on a family of admissible G-covers. This formula can be interpreted as a representation-ringvalued relative Riemann-Hurwitz formula for families of admissible G-covers. Contents

Abstract. For any variety X with an action of a split torus over a field, we establish a spectral sequence connecting the equivariant and the non-equivariant higher Chow groups of X. For X smooth and projective, we give an explicit formula relating the equivariant and the non-equivariant higher Chow groups. This in particular implies that the spectral sequence degenerates. These results also hold with rational coefficients for action of any connected and split reductive group. As applications to the above results, we show that for a connected linear algebraic group G acting on a smooth projective variety X, the forgetful map K G i (X) → Ki(X) induces an isomorphism K G i (X)/IGK G i (X) ∼ = − → Ki(X) with rational coefficients. This generalizes a result of Graham to higher K-theory of such varieties. We also prove a refinement of the equivariant Riemann-Roch theorem of [17] and [16] for such varieties, which leads to the generalizations of the main results of [7] and [9] to the equivariant higher K-theory. 1.

Abstract. We prove the conjectures of Graham–Kumar [GrKu08] and Griffeth– Ram [GrRa04] concerning the alternation of signs in the structure constants for torus-equivariant K-theory of generalized flag varieties G/P. These results are immediate consequences of an equivariant homological Kleiman transversality principle for the Borel mixing spaces of homogeneous spaces, and their subvarieties, under a natural group action with finitely many orbits. The computation of the coefficients in the expansion of the equivariant K-class of a subvariety in terms of Schubert classes is reduced to an Euler characteristic using the homological transversality theorem for non-transitive group actions due to S. Sierra. A vanishing theorem, when the subvariety has rational singularities, shows that the Euler characteristic is a sum of at most one term—the top one—with a well-defined sign. The vanishing is proved by suitably modifying a geometric argument due to M. Brion in ordinary K-theory that brings Kawamata–Viehweg vanishing to bear. 1.