Chern classes of birational varieties

Paolo Aluffi

Let $\varphi: V\dashrightarrow W$ be a birational map between smooth algebraic varieties which does not change the canonical class (in the sense of Batyrev). We prove that the total homology Chern classes of $V$ and $W$ are push-forwards of the same class from a resolution of indeterminacies of $\varphi$.

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.