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Shimura data and corners: topology

J. Wildeshaus

TL;DR

The paper addresses relating the Borel–Serre and Baily–Borel compactifications of a Shimura variety by constructing and analyzing a direct map $p: M^{BS}\to M^*$ that respects boundary stratifications. It develops a Shimura-data–driven framework, introducing spaces of type $S'$ and a concrete geodesic action described by weight co-characters, to build the corner-type manifold ${\mathfrak X}^{BS}$ and its adelic counterpart $M^{BS}$. It proves that the map $p$ is continuous and interacts transparently with the stratifications, and describes fibers and local splittings; when the level is neat, $p^K$ is a locally trivial stratified fibration over canonical strata of $M^*$. The approach emphasizes a systematic use of Shimura data to control the boundary geometry without relying on Langlands decompositions, with potential applications to cohomology via localization and boundary analysis.

Abstract

The purpose of this article is to give a new construction of the map relating the Borel-Serre and the Baily-Borel compactifications of a Shimura variety (Zucker 1983), and to provide a close analysis of its main properties.

Shimura data and corners: topology

TL;DR

The paper addresses relating the Borel–Serre and Baily–Borel compactifications of a Shimura variety by constructing and analyzing a direct map that respects boundary stratifications. It develops a Shimura-data–driven framework, introducing spaces of type and a concrete geodesic action described by weight co-characters, to build the corner-type manifold and its adelic counterpart . It proves that the map is continuous and interacts transparently with the stratifications, and describes fibers and local splittings; when the level is neat, is a locally trivial stratified fibration over canonical strata of . The approach emphasizes a systematic use of Shimura data to control the boundary geometry without relying on Langlands decompositions, with potential applications to cohomology via localization and boundary analysis.

Abstract

The purpose of this article is to give a new construction of the map relating the Borel-Serre and the Baily-Borel compactifications of a Shimura variety (Zucker 1983), and to provide a close analysis of its main properties.
Paper Structure (8 sections, 508 equations)

This paper contains 8 sections, 508 equations.

Table of Contents

  1. Introduction
  2. Homogeneous spaces of type $S'$
  3. Geodesic action
  4. The manifold with corners ${\mathfrak{X}}^{BS}$
  5. The continuous map $p$ from the Borel--Serre to the Baily--Borel compactification
  6. The canonical stratifications
  7. The fibres of the map $p$
  8. Triviality of $p$ over each stratum