Coherent cohomology of Shimura varieties and automorphic forms.

We show that the cohomology of canonical extensions of automorphic vector bundles over toroidal compactifications of Shimura varieties can be computed by relative Lie algebra cohomology of automorphic forms. Our result is inspired by and parallel to Borel and Frankeu0027s work on the cohomology of automorphic local systems on locally symmetric spaces, and also generalizes a theorem of Mumford.