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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 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 URI: http://hdl.handle.net/2142/26229 Rights Information: Copyright 2011 Artan Sheshmani Date Available in IDEALS: 2011-08-25 Date Deposited: 2011-08
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