Perverse sheaves and the reductive Borel-Serre compactification

TL;DR

This paper explores the theory of perverse sheaves on the reductive Borel-Serre compactification of Hermitian locally symmetric spaces, aiming to connect topological and algebraic compactifications for advanced geometric analysis.

Contribution

It introduces the application of perverse sheaves to the reductive Borel-Serre compactification and demonstrates the calculation of extensions, advancing the understanding of their topological and algebraic interplay.

Findings

Decomposition theorem holds for the key map between compactifications.

Methods to compute extensions of simple perverse sheaves are developed.

Illustrative example with Sp(4,R) demonstrates the approach.

Abstract

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 sheaves on the reductive Borel-Serre compactification and illustrate with the example of Sp(4,R).