Towards studying of the higher rank theory of stable pairs

Sheshmani, Artan

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

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

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

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

Copyright and License Information

Copyright 2011 Artan Sheshmani

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