Categorification of the Kirwan map

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois Urbana-Champaign

Year dc:date

2025

Author dc:creator

• O'Neill, Ciaran

Contributors dc:contributor

• Dodd, Christopher
• Katz, Sheldon
• Pascaleff, James
• Berwick-Evans, Daniel

• Kirwan surjectivity
• Hochshild homology
• D-modules,

Statement dc:rights

• Copyright 2025 Ciaran O'Neill

Language dc:language

en

Handle dc:identifier

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

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/132574