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

The L-2-cohomology of an arithmetic quotient of an Hermitian symmetric spac e (i.e., of a locally symmetric variety) is known to have the topological i nterpretation as the intersection homology of its Baily-Borel Satake compac tification. In this article, we observe that even without the Hermitian hyp othesis, the LP-cohomology of an arithmetic quotient, for p finite and suff iciently large, is isomorphic to the ordinary cohomology of its reductive B orel-Serre compactification. We use this to generalize a theorem of Mumford concerning homogeneous vector bundles, their invariant Chem forms and the canonical extensions of the bundles; here, though, we are referring to cano nical extensions to the reductive Borel-Serre compactification. of any arit hmetic quotient. To achieve that, we give a systematic discussion of vector bundles and Chem classes on stratified spaces.