ABSTRACT. Motivic characteristic classes of possibly singular algebraic varieties are homology class versions of motivic characteristics, not classes in the so-called motivic (co)homology. This paper is a survey of them, with emphasis on capturing infinitude finitely and on the motivic nature, in other words, the scissor relation or additivity.

1. Introduction

Characteristic classes are usually cohomological objects defined on real or complex vector bundles. For a smooth manifold, for instance, its characteristic classes are defined through the tangent bundle. For real vector bundles, Stiefel— Whitney classes and Pontraygin classes are fundamental; for complex vector bundles, the Chern class is the fundamental one.

When it comes to a non-manifold space, such as a singular real or complex algebraic or analytic variety, one cannot talk about its cohomological characteristic class, unlike the smooth case, because one cannot define its tangent bundle—although one can define some reasonable substitutes, such as the tangent cone and tangent star cone, which are not vector bundles, but stratified vector bundles.