  Let X be an irreducible complex variety, S a stratification of X and F a
holomorphic vector bundle on the open statum. We give geometric conditions on S
and F that produce a natural extension of the k-th Chern class F as a class in
the complex cohomology of X of Hodge level at least k. When X is the
Baily-Borel compactification of a locally symmetric variety with its
stratification by boundary components, and F an automorphic bundle on its
interior, then this recovers and refines a theorem of Goresky-Pardon. In
passing we define a class of simplicial resolutions of the Baily-Borel
compactification that can be used to define its mixed Hodge structure. Finally,
we use the Beilinson regulator for the rationals to show that when X is the
Satake (=Baily-Borel) compactification of A_g and F the Hodge bundle (with g
large compared to k), the Goresky-Pardon k-th Chern class extension has nonzero
imaginary part and gives rise to a basic Tate extension. This also shows that
the lifts of Chern classes constructed by Goresky and Pardon need not be real
and thus answers (negatively) a question asked by these authors.
