Interpolation of characteristic classes of singular hypersurfaces

P. Aluffi, J.-P. Brasselet

We show that the Chern-Schwartz-MacPherson class of a hypersurface
*X* in a nonsingular variety *M* `interpolates' between two
other notions of characteristic classes for singular varieties,
provided that the singular locus of *X* is smooth and that certain
numerical invariants of *X* are constant along this locus. This
allows us to define a lift of the Chern-Schwartz-MacPherson class of
such `nice' hypersurfaces to intersection homology. As another
application, the interpolation result leads to an explicit formula for
the Chern-Schwartz-MacPherson class of *X* in terms of its polar
classes.