  In this paper we study some new theories of characteristic homology classes
for singular complex algebraic varieties. First we introduce a natural
transformation T_{y}: K_{0}(var/X) -> H_{*}(X,Q)[y] commuting with proper
pushdown, which generalizes the corresponding Hirzebruch characteristic. 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. 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) and the Todd class transformation in the singular
Riemann-Roch theorem of Baum-Fulton-MacPherson (for y=0). In fact, T_{y} is the
composition of a generalized version of this Todd class transformation due to
Yokura, and a new 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 G_{0}(X) is the Grothendieck group of coherent
sheaves, and the construction of mC_{*} is based on some results from the
theory of algebraic mixed Hodge modules due to M.Saito. In the final part of
the paper we use the algebraic cobordism theory of Levine and Morel to lift
mC_{*} further up to a natural transformation mC'_{*} from K_{0}(var/X) to a
suitable universal Borel-Moore weak homology theory. Moreover, all our results
can be extended to varieties over a base field k, which can be embedded into
the complex numbers.
