University of Illinois Urbana-Champaign

Towards studying of the higher rank theory of stable pairs

Sheshmani, Artan

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Sheshmani_Artan.pdf
Thesis-final-draft-2011-0620.bbl

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https://hdl.handle.net/2142/26229

Description

Title
Towards studying of the higher rank theory of stable pairs
Author(s)
Sheshmani, Artan
Issue Date
2011-08-25T22:19:38Z
Director of Research (if dissertation) or Advisor (if thesis)
  • Katz, Sheldon
  • Nevins, Thomas A.
Doctoral Committee Chair(s)
Bradlow, Steven B.
Committee Member(s)
  • Katz, Sheldon
  • Nevins, Thomas A.
  • Schenck, Henry K.
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
2011-08-25T22:19:38Z
Keyword(s)
  • Calabi-Yau threefold
  • Stable pairs
  • Deformation-obstruction theory
  • Derived categories
  • Equivariant cohomology
  • Virtual localization
  • Wallcrossing
Abstract
This thesis is composed of two parts. In the first part we introduce a higher rank analog of the Pandharipande-Thomas theory of stable pairs on a Calabi-Yau threefold $X$. More precisely, we develop a moduli theory for frozen triples given by the data $\mathcal{O}_X^{\oplus r}(-n)\xrightarrow{\phi} F$ where $F$ is a sheaf of pure dimension $1$. The moduli space of such objects does not naturally determine an enumerative theory: that is, it does not naturally possess a perfect symmetric obstruction theory. Instead, we build a zero-dimensional virtual fundamental class by hand, by truncating a deformation-obstruction theory coming from the moduli of objects in the derived category of $X$. This yields the first deformation-theoretic construction of a higher-rank enumerative theory for Calabi-Yau threefolds. We calculate this enumerative theory for local $\mathbb{P}^1$ using the Graber-Pandharipande virtual localization technique. In the second part of the thesis we compute the Donaldson-Thomas type invariants associated to frozen triples using the wall-crossing formula of Joyce-Song and Kontsevich-Soibelman.
Graduation Semester
2011-08
Permalink
http://hdl.handle.net/2142/26229
Copyright and License Information
Copyright 2011 Artan Sheshmani

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