On the reductive Borel-Serre compactification: Lp-cohomology of arithmetic groups (for large p)

The L²-cohomology of an arithmetic quotient of an Hermitian symmetric space (i.e., of a locally symmetric variety) is known to have the topological interpretation as the intersection homology of its Baily-Borel Satake compactification. In this article, we observe that even without the Hermitian hypothesis, the L^p-cohomology of an arithmetic quotient, for p finite and sufficiently large, is isomorphic to the ordinary cohomology of its reductive Borel-Serre compactification. We use this to generalize a theorem of Mumford concerning homogeneous vector bundles, their invariant Chern forms and the canonical extensions of the bundles; here, though, we are referring to canonical extensions to the reductive Borel-Serre compactification of any arithmetic quotient. To achieve that, we give a systematic discussion of vector bundles and Chern classes on stratified spaces.