Gopakumar-Vafa invariant and Macdonald formula

In this thesis, I will introduce the Gopakumar-Vafa(GV) invariant and show one calculation on the nonreduced cycle. The GV invariant is an integral invariant predicted by physicist that counts the number of curves inside a given Calabi-Yau threefold. The definition has been conjectured by Maulik-Toda in 2016 in terms of perverse sheaf. I will use this definition on the total space of canonical bundle of P2 and compute the associated invariants. I will introduce a Gopakumar-Vafa/Pandharipande-Thomas correspondence on the level of perverse sheaves, inspired by the work of Migliorini-Shende-Viviani. I will verify that my calculation actually proves part of the conjecture. I have shown a strong evidence for this conjecture in the case of degree 2.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Zhao, Lutian

Contributors dc:contributor

• Katz, Sheldon
• Bradlow, Steven
• Dodd, Christopher
• Haboush, William

• Moduli spaces, Gopakumar-Vafa invariants, Macdonald Formula, Hilbert Scheme, Decomposition Theorem

Statement dc:rights

• Copyright 2021, Lutian Zhao

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/113016

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/113016