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On the reductive Borel-Serre compactification, II: Excentric quotients and least common modifications
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 130, Number 4, August 2008
- pp. 859-912
- 10.1353/ajm.0.0010
- Article
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Let $X$ be a locally symmetric variety, i.e., the quotient of a bounded symmetric domain by a (say) neat arithmetically-defined group of isometries. Let ${\overline X}^{\rm exc}$ and $X^{{\rm tor,exc}}$ denote its excentric Borel-Serre and toroidal compactifications respectively. We determine their least common modification and use it to prove a conjecture of Goresky and Tai concerning canonical extensions of homogeneous vector bundles. In the process, we see that ${\overline X}^{\rm exc}$ and $X^{{\rm tor,exc}}$ are homotopy equivalent.