Weighted Cohomology, Hodge Theory and Intersection Cohomology of Shimura varieties

TL;DR

This paper establishes a deep connection between the intersection cohomology of Shimura varieties and mixed Hodge structures, providing new tools for understanding their geometric and topological properties.

Contribution

It proves an identification between intersection cohomology and top weight quotients of mixed Hodge structures, introducing canonical cycle classes and functorial operations.

Findings

Identification of intersection cohomology with top weight quotients

Introduction of canonical cycle classes for special cycles

Relation of geometric volumes to topological terms

Abstract

We prove that the intersection cohomology of the Baily-Borel compactification of a complex Shimura variety is identified with the top weight quotient of the mixed Hodge structure on the reductive Borel-Serre compactification. This yields canonical cup products and functorial pullbacks on the intersection cohomology. As an application, we introduce canonical cycle classes associated to special cycles, relating analytic geometric volumes of non-compact Shimura varieties to topological terms.