Hirzebruch Classes and Motivic Chern Classes for Singular Spaces

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 transformationmC y : K 0 ( var /X) → G 0 (X) ⊗ ℤ[y], which generalizes the total λ-class λ y (T*X) 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 [Formula: see text]-modules. A first construction of mC y is based on resolution of singularities and a suitable "blow-up" relation, following the work of Du Bois, Guillén, Navarro Aznar, Looijenga and Bittner. A second more functorial construction of mC y is based on some results from the theory of algebraic mixed Hodge modules due to M. Saito. 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 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.