On the reductive Borel-Serre compactification, II: Excentric quotients and least common modifications

Abstract

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.