Categorification of the Kirwan map

O'Neill, Ciaran

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https://hdl.handle.net/2142/132574 Copy

Description

Title

Categorification of the Kirwan map

Author(s)

O'Neill, Ciaran

Issue Date

2025-12-04

Director of Research (if dissertation) or Advisor (if thesis)

Dodd, Christopher

Doctoral Committee Chair(s)

Katz, Sheldon

Committee Member(s)

• Pascaleff, James
• Berwick-Evans, Daniel

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Kirwan surjectivity
• Hochshild homology
• D-modules,

Abstract

Given a Hamiltonian action of a group on a scheme we can consider the Hamiltonian reduction. This furnishes us with a notion of quotient. From the construction it follows that we have a map from the equivariant cohomology of the scheme to the cohomology of the quotient. It is natural to ask if this map is surjective. This question is known as Kirwan surjectivity. There is much literature surrounding this question but the exact boundary of when Kirwan surjectivity holds is unknown. The document that follows explains a method for approaching the closely related question of Borel-Moore Kirwan surjectivity. The approach relies on deep results from derived algebraic geometry. Along the way it is shown that the equivariant category of D-modules with certain support condition is equivalent to the category of D-modules on the quotient. It is then shown that we can associate particular equivariant D-modules to characters of the group and these equivariant D-modules are compatible in a natural sense with the well studied equivariant O-modules associated to characters. This gives a framework for an approach to Borel-Moore Kirwan surjectivity. A result of McBreen and Webster immediately allows us to realise that Borel-Moore Kirwan surjectivity holds in the case of hypertorics.

Graduation Semester

2025-12

Type of Resource

Thesis

Handle URL

https://hdl.handle.net/2142/132574

Copyright and License Information

Copyright 2025 Ciaran O'Neill

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