Current Developments in Mathematics

On the Cohomology of Locally Symmetric Spaces and of their Compactifications

Leslie Saper

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This expository article gives an introduction to the (generalized) conjecture of Rapoport and Goresky-MacPherson which identifies the intersection cohomology of a real equal-rank Satake compactification of a locally symmetric space with that of the reductive Borel-Serre compactification. We motivate the conjecture with examples and then give an introduction to the various topics that are involved: intersection cohomology, the derived category, and compactifications of a locally symmetric space, particularly those above. We then give an overview of the theory of L-modules and micro-support which was developed to solve the conjecture but has other important applications as well. We end with sketches of the proofs of three main theorems on L-modules that lead to the resolution of the conjecture. The text is enriched with many examples, illustrations, and references to the literature.

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Current Developments in Mathematics Volume 2002 (2002), 219-289.

First available in Project Euclid: 29 June 2004

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Saper , Leslie. On the Cohomology of Locally Symmetric Spaces and of their Compactifications. Current Developments in Mathematics 2002 (2002), 219--289.

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