In this paper we study some new theories of characteristic homology classes of singular complex algebraic (or compactifiable analytic) spaces. We introduce a motivic Chern class transformationmCy: K0(var/X) → G0(X) ⊗ ℤ[y], which generalizes the total λ-class λy(T*X) of the cotangent bundle to singular spaces. Here K0(var/X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration, and G0(X) is the Grothendieck group of coherent sheaves of -modules. A first construction of mCy is based on resolution of singularities and a suitable "blow-up" relation, following the work of Du Bois, Guillen, Navarro Aznar, Looijenga and Bittner. A second more functorial construction of mCy is based on some results from the theory of algebraic mixed Hodge modules due to M. Saito. We define a natural transformation Ty* : K0(var/X) → H*(X) ⊗ ℚ[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. Ty* is a homology class version of the motivic measure corresponding to a suitable specialization of the well-known Hodge polynomial. This transformation unifies the Chern class transformation of MacPherson and Schwartz (for y = -1), the Todd class transformation in the singular Riemann-Roch theorem of Baum–Fulton–MacPherson (for y = 0) and the L-class transformation of Cappell-Shaneson (for y = 1). We also explain the relation among the "stringy version" of our characteristic classes, the elliptic class of Borisov–Libgober and the stringy Chern classes of Aluffi and De Fernex–Lupercio–Nevins–Uribe. All our results can be extended to varieties over a base field k of characteristic 0.

Hirzebruch classes and motivic chern classes for singular spaces

In this paper we study some new theories of characteristic homology classes for singular complex algebraic (or compactifiable analytic) spaces. We introduce a motivic Chern class transformation mC_{*}: K_{0}(var/X)-> G_{0}(X)[y], which generalizes the total \lambda-class of the cotangent bundle to singular spaces. Here K_{0}(var/X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration, and G_{0}(X) is the Grothendieck group of coherent sheaves of O_{X}-modules. We define a natural transformation T_{y*}: K_{0}(var/X)-> H_{*}(X,Q)[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. T_{y*} is a homology class version of the motivic measure corresponding to suitable specialization of the well known Hodge polynomial. This transformation unifies the Chern class transformation of MacPherson and Schwartz (for y=-1), the Todd class transformation of the singular Riemann-Roch theorem of Baum-Fulton-MacPherson (for y=0) and the L-class transformation of Cappell-Shaneson (for y=1). In the simplest case of a normal Gorenstein variety with ``canonical singularities'' we also explain a relation among the ``stringy version'' of our characteristic classses, the elliptic class of Borisov-Libgober and the stringy Chern classes of Aluffi and De Fernex-Lupercio-Nevins-Uribe. Moreover, all our results can be extended to varieties over a base field k of characteristic 0.

Hirzebruch classes and motivic Chern classes for singular spaces

Symmetric operations in algebraic cobordism

We define and study Gysin morphisms on mixed motives over a perfect field. Our construction extends the case of closed immersions, already known from results of Voevodsky, to arbitrary projective morphisms. We prove several classical formulas in this context, such as the projection and excess intersection formulas, and some more original ones involving residues. Finally, we give an application of this construction to duality and motive with compact support.

Around the Gysin triangle I

Chow-Witt groups and Grothendieck-Witt groups of regular schemes

Pairings in triangular Witt theory

http://www.crm.umontreal.ca/K/pdf/zainoulline.pdf

Oriented cohomology and motivic decompositions of relative cellular spaces

Gysin maps in Balmer–Witt theory

Formulas for the derived Witt groups of projective bundles are obtained. We deduce them from general properties of Witt theory, with the help of twisted Thom isomorphisms, avoiding explicit use of triangulated categories. Witt groups of completely split quadrics are also considered.

Alexander Nenashev

Papers

Pairings in triangular Witt theory

Oriented cohomology and motivic decompositions of relative cellular spaces

Gysin maps in Balmer–Witt theory

On the Witt groups of projective bundles and split quadrics: geometric reasoning

Gysin maps in oriented theories