HIRZEBRUCH CLASSES AND MOTIVIC CHERN CLASSES FOR SINGULAR SPACES

<jats:p>In this paper we study some new theories of characteristic homology classes of singular complex algebraic (or compactifiable analytic) spaces.</jats:p><jats:p>We introduce a motivic Chern class transformationmC<jats:sub>y</jats:sub>: K<jats:sub>0</jats:sub>( var /X) → G<jats:sub>0</jats:sub>(X) ⊗ ℤ[y], which generalizes the total λ-class λ<jats:sub>y</jats:sub>(T*X) of the cotangent bundle to singular spaces. Here K<jats:sub>0</jats:sub>( 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<jats:sub>0</jats:sub>(X) is the Grothendieck group of coherent sheaves of [Formula: see text]-modules. A first construction of mC<jats:sub>y</jats:sub>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<jats:sub>y</jats:sub>is based on some results from the theory of algebraic mixed Hodge modules due to M. Saito.</jats:p><jats:p>We define a natural transformation T<jats:sub>y*</jats:sub>: K<jats:sub>0</jats:sub>( var /X) → H<jats:sub>*</jats:sub>(X) ⊗ ℚ[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. T<jats:sub>y*</jats:sub>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).</jats:p><jats:p>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.</jats:p><jats:p>All our results can be extended to varieties over a base field k of characteristic 0.</jats:p>