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27
A Survey of Characteristic Classes of Singular Spaces
"... 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 va ..."
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Cited by 22 (4 self)
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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 Lclass. 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.
POSITIVITY AND KLEIMAN TRANSVERSALITY IN EQUIVARIANT KTHEORY OF HOMOGENEOUS SPACES
"... Abstract. We prove the conjectures of Graham–Kumar [GrKu08] and Griffeth– Ram [GrRa04] concerning the alternation of signs in the structure constants for torusequivariant Ktheory of generalized flag varieties G/P. These results are immediate consequences of an equivariant homological Kleiman trans ..."
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Abstract. We prove the conjectures of Graham–Kumar [GrKu08] and Griffeth– Ram [GrRa04] concerning the alternation of signs in the structure constants for torusequivariant Ktheory 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 Kclass of a subvariety in terms of Schubert classes is reduced to an Euler characteristic using the homological transversality theorem for nontransitive 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 welldefined sign. The vanishing is proved by suitably modifying a geometric argument due to M. Brion in ordinary Ktheory that brings Kawamata–Viehweg vanishing to bear. 1.
QUANTUM KTHEORY OF GRASSMANNIANS
, 2008
"... We show that (equivariant) Ktheoretic 3point GromovWitten invariants of genus zero on a Grassmann variety are equal to triple intersections computed in the (equivariant) Ktheory of a twostep flag manifold, thus generalizing an earlier result of Buch, Kresch, and Tamvakis. In the process we sho ..."
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Cited by 14 (6 self)
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We show that (equivariant) Ktheoretic 3point GromovWitten invariants of genus zero on a Grassmann variety are equal to triple intersections computed in the (equivariant) Ktheory of a twostep flag manifold, thus generalizing an earlier result of Buch, Kresch, and Tamvakis. In the process we show that the GromovWitten variety of curves passing through 3 general points is irreducible and rational. Our applications include Pieri and Giambelli formulas for the quantum Ktheory ring of a Grassmannian, which determine the multiplication in this ring. Our formula for GromovWitten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of Chaput, Manivel, and Perrin.
Equivariant Todd classes for toric varieties
"... 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 RiemannRoch theorem and the computation of the equivariant cohomology and equivariant homology of toric varieties. This ..."
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Cited by 10 (4 self)
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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 RiemannRoch theorem and the computation of the equivariant cohomology and equivariant homology of toric varieties. This research was supported in part by NSF grant DMS9504522 and DMS9803593
Equivariant Cobordism of Schemes
 DOCUMENTA MATH.
, 2012
"... Let k be a field of characteristic zero. For a linear algebraic group G over k acting on a scheme X, we define the equivariant algebraic cobordism of X and establish its basic properties. We explicitly describe the relation of equivariant cobordism with equivariant Chow groups, Kgroups and complex ..."
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Cited by 8 (4 self)
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Let k be a field of characteristic zero. For a linear algebraic group G over k acting on a scheme X, we define the equivariant algebraic cobordism of X and establish its basic properties. We explicitly describe the relation of equivariant cobordism with equivariant Chow groups, Kgroups and complex cobordism. We show that the rational equivariant cobordism of a Gscheme can be expressed as the Weyl group invariants of the equivariant cobordism for the action of a maximal torus of G. As applications, we show that the rational algebraic cobordism of the classifying space of a complex linear algebraic group is isomorphic to its complex cobordism.
On the birational geometry of the universal Picard variety
 International Math. Res. Notices
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RiemannRoch for equivariant Ktheory
, 2009
"... Abstract. The goal of this paper is to prove the equivariant version of Bloch’s RiemannRoch isomorphism between the higher algebraic Ktheory 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 characte ..."
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Cited by 8 (8 self)
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Abstract. The goal of this paper is to prove the equivariant version of Bloch’s RiemannRoch isomorphism between the higher algebraic Ktheory 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 Kgroups 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 RiemannRoch isomorphism ̂ KG G bτ X i (X) −− → ̂ CH ∗ G (X, i). The case i = 0 provides a stronger form of the RiemannRoch theorem of Edidin and Graham (cf. [6]). 1.
Riemann–Roch for quotients and Todd classes of simplicial toric varieties
 Commun. Algebra
"... Abstract. In this paper we give an explicit formula for the RiemannRoch 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 f ..."
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Cited by 5 (1 self)
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Abstract. In this paper we give an explicit formula for the RiemannRoch 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.
EQUIVARIANT KTHEORY AND HIGHER CHOW GROUPS OF SMOOTH VARIETIES
, 2009
"... For any variety X with an action of a split torus over a field, we establish a spectral sequence connecting the equivariant and the nonequivariant higher Chow groups of X. For X smooth and projective, we give an explicit formula relating the equivariant and the nonequivariant higher Chow groups. T ..."
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Cited by 5 (5 self)
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For any variety X with an action of a split torus over a field, we establish a spectral sequence connecting the equivariant and the nonequivariant higher Chow groups of X. For X smooth and projective, we give an explicit formula relating the equivariant and the nonequivariant 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 Ktheory of such varieties. We also prove a refinement of the equivariant RiemannRoch 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 Ktheory.