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

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

2007-04-11

arxiv.org/abs/0704.1335v1

Amer. J. Math. 123 (2001), 951-984

Mathematics

Algebraic Geometry

32 pages

Scientific paper

The $L^2$-cohomology 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

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