The stratified homotopy type of the reductive Borel-Serre compactification

Perverse sheaves and the reductive Borel-Serre compactification

We briefly introduce the theory of perverse sheaves with special attention to the topological situation where strata can have odd dimension. This is part of a project to use perverse sheaves on the topological reductive Borel-Serre compactification of a Hermitian locally symmetric space as a tool to study perverse sheaves on the Baily-Borel compactification, a projective algebraic variety. We sketch why the decomposition theorem holds for the natural map between the reductive Borel-Serre and the Baily-Borel compactifications. We demonstrate how to calculate extensions of simple perverse sheave…

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

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 …

环路群的Borel–Serre型加边与典范对

Borel--Serre type bordifications and Canonical Pairs for Loop groups

本文为实环路群对称空间构造Borel–Serre型加边，证明其边界同伦于仿射有理Tits building，并借助Garland约化理论、半稳定性概念研究算术群作用的商空间及其非紧致性。

Bianchi's additional symmetries

In a 2012 note in Comptes Rendus Math{é}matique, the author did try to answer a question of Jean-Pierre Serre; it has recently been announced that the scope of that answer needs an adjustment, and the details of this adjustment are given in the present paper. The original question is the following. Consider the ring of integers O in an imaginary quadratic number field, and the Borel-Serre compactification of the quotient of hyperbolic 3-space by SL 2 (O). Consider the map $α$ induced on homology when attaching the boundary into the Borel-Serre compactification. How can one determine the kernel…

Shimura 数据与角点：拓扑

Shimura data and corners: topology

本文给出 Shimura 簇的 Borel-Serre 紧化与 Baily-Borel 紧化之间映射的新构造，并分析其主要性质。