A canonical lift of Chern-Mather classes to intersection homology

Jean-Paul Brasselet, Andrzej Weber

17 pages

There are several ways to generalize characteristic classes for singular algebraic varieties. The simplest ones to describe are Chern-Mather classes obtained by Nash blow up. They serve as an ingredient to construct Chern-MacPherson-Schwartz classes. Unfortunately, they all are defined in homology. There are examples showing, that they do not lie in the image of Poincar\'e morphism. On the other hand they are represented by an algebraic cycles. Barthel, Brasselet, Fiesler, Kaup and Gabber have shown that, any algebraic cycle can be lifted to intersection homology. Nevertheless, a lift is not unique. The Chern-Mather classes are represented by polar varieties. We show how to define a canonical lift of Chern-Mather classes to intersection homology. Instead of the polar variety alone, we consider it as a term in the whole sequence of inclusions of polar varieties. The inclusions are of codimension one. In this case the lifts are unique.