University of Illinois Urbana-Champaign

Gopakumar-Vafa invariant and Macdonald formula

Zhao, Lutian

Content Files
ZHAO-DISSERTATION-2021.pdf

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

Description

Title
Gopakumar-Vafa invariant and Macdonald formula
Author(s)
Zhao, Lutian
Issue Date
2021-07-14
Director of Research (if dissertation) or Advisor (if thesis)
Katz, Sheldon
Doctoral Committee Chair(s)
Bradlow, Steven
Committee Member(s)
  • Dodd, Christopher
  • Haboush, William
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2022-01-12T21:45:36Z
Keyword(s)
Moduli spaces, Gopakumar-Vafa invariants, Macdonald Formula, Hilbert Scheme, Decomposition Theorem
Abstract
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.
Graduation Semester
2021-08
Type of Resource
Thesis
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http://hdl.handle.net/2142/113016
Copyright and License Information
Copyright 2021, Lutian Zhao

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