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Title:Towards studying of the higher rank theory of stable pairs
Author(s):Sheshmani, Artan
Director of Research:Katz, Sheldon; Nevins, Thomas A.
Doctoral Committee Chair(s):Bradlow, Steven B.
Doctoral Committee Member(s):Katz, Sheldon; Nevins, Thomas A.; Schenck, Henry K.
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):Calabi-Yau threefold
Stable pairs
Deformation-obstruction theory
Derived categories
Equivariant cohomology
Virtual localization
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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