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Weighted Cohomology, Hodge Theory and Intersection Cohomology of Shimura varieties

Mingyu Ni

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.

Weighted Cohomology, Hodge Theory and Intersection Cohomology of Shimura varieties

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.

Paper Structure

This paper contains 16 sections, 26 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalities on Hodge Theory
  4. Weight Truncations and Weightless Cohomology
  5. Parabolic Subgroups and Weight Profiles
  6. Special Forms and Weighted Cohomology
  7. Analytic Aspects of Weighted Cohomology
  8. Borel's Integrability Criterion
  9. Integrability of Special Forms
  10. An Eventual Quasi-isomorphism and Surjectivity
  11. Weight Comparison and Intersection Cohomology
  12. Weight Comparison
  13. Operations on Intersection Cohomology
  14. Further Applications
  15. Automorphic Lefschetz Properties for Intersection Cohomology
  16. ...and 1 more sections

Key Result

Theorem 1.1

Theorems & Definitions (56)

  • Theorem 1.1
  • Example 2.1
  • Proposition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • Definition 2.7
  • Example 2.8
  • ...and 46 more