Local forms for the double Aₙ quiver and Gopakuma–Vafa invariants

Zhang, Hao (2025) Local forms for the double Aₙ quiver and Gopakuma–Vafa invariants. PhD thesis, University of Glasgow.

Abstract

This thesis investigates the crepant (partial) resolutions of cAn singularities and their associated Gopakumar–Vafa (GV) invariants via noncommutative contraction algebras.

We begin in Chapter 3 by generalising GV invariants to crepant partial resolutions of cAₙ singularities and demonstrate that these generalised invariants satisfy Toda’s formula. Furthermore, we prove that generalised GV invariants are determined by the isomorphism class of the contraction algebra.

In Chapter 4 we focus on crepant resolutions of cAₙ singularities, and introduce several intrinsic definitions of a Type A potential on the doubled Aₙ quiver Qₙ, which includes a single loop at each vertex. Through applying coordinate changes, we then:

(1) Via monomialization, expresses these potentials in a particularly nice form;
(2) Show that Type A potentials classify crepant resolutions of cAₙ singularities;
(3) Confirm the Realisation Conjecture of Brown–Wemyss within this context.

We also provide an example of a non-isolated cA₂ singularity which illustrates that the Donovan–Wemyss Conjecture fails for non-isolated cDV singularities.

Building upon the correspondence between crepant resolutions of cAₙ singularities and monomialized Type A potentials, in Chapter 5 we:

(1) Introduce a filtration structure on the parameter space of monomialized Type A potentials with respect to the generalised GV invariants;
(2) Derive numerical constraints on the possible tuples of GV invariants, and explicitly classify all tuples arising from crepant resolutions of cA₂ singularities.

For n ≤ 3, in Chapter 6 we further provide a complete classification of Type A potentials (without loops) up to isomorphism, as well as a classification of those with finite-dimensional Jacobi algebras up to derived equivalence. These results yield various algebraic consequences, including applications to certain tame algebras of quaternion type studied by Erdmann, for which we describe all basic algebras within the derived equivalence class.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Additional Information: Supported by funding from the China Scholarship Council.
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics
• Funder's Name: China Scholarship Council
• Supervisor's Name: Wemyss, Professor Michael
• Date of Award: 2025
• Depositing User: Theses Team
• Unique ID: glathesis:2025-85553
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 27 Oct 2025 16:19
• Last Modified: 28 Oct 2025 14:49
• Thesis DOI: 10.5525/gla.thesis.85553
• URI: https://theses.gla.ac.uk/id/eprint/85553