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Towards studying of the higher rank theory of stable pairs

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Title: Towards studying of the higher rank theory of stable pairs
Author(s): Sheshmani, Artan
Director of Research: Katz, Sheldon H.; Nevins, Thomas A.
Doctoral Committee Chair(s): Bradlow, Steven B.
Doctoral Committee Member(s): Katz, Sheldon H.; Nevins, Thomas A.; Schenck, Henry K.
Department / Program: Mathematics
Discipline: Mathematics
Degree Granting Institution: University of Illinois at Urbana-Champaign
Degree: Ph.D.
Genre: Dissertation
Subject(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.
Issue Date: 2011-08-25
Rights Information: Copyright 2011 Artan Sheshmani
Date Available in IDEALS: 2011-08-25
Date Deposited: 2011-08

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