  Let X be a Hermitian locally symmetric space. We prove that every Chern class
of X has a canonical lift to the cohomology of the Baily- Borel-Satake
compactification X* of X and that the resulting Chern numbers satisfy the
Hirzebruch proportionality formula with respect to the compact dual X^ of X.
The same result holds for any automorphic vector bundle over X in place of the
tangent bundle. As a consequence there is a surjection of the subalgebra of
H*(X*) generated by these lifted classes onto H*(X^). The method of proof is to
construct fiberwise flat connections on these bundles near the singular strata
of X*, where one then finds de Rham representatives of the Chern classes which
are pulled back from the strata.
