2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74604In 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 λ-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 0.55 pages, some typos corrected, this is a completely new and improved version of the paper math.AG/0405412Algebraic GeometryAlgebraic TopologyK-Theory and Homology14C17;19L10;32S20;55R40text