  A theorem of Batyrev's asserts that if two nonsingular varieties V,W are
birational, and their canonical bundles agree after pull-back to a resolution
of indeterminacies of a birational map between them, then the Betti numbers of
V and W coincide. We prove that, in the same hypotheses, the total homology
Chern classes of V and W are push-forwards of the same class in the Chow group
of the resolution.
  For example, it follows that the push-forward of the total Chern class of a
crepant resolution of a singular variety is independent of the resolution.
