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.