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1. Sheshmani, Artan. Towards studying of the higher rank theory of stable pairs.

Degree: PhD, 0439, 2011, University of Illinois – Urbana-Champaign

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

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 𝓞_{X}^{⊕ r}(-n)\xrightarrow{φ} 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 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.
*Advisors/Committee Members: Katz, Sheldon (advisor), Nevins, Thomas A. (advisor), Bradlow, Steven B. (Committee Chair), Katz, Sheldon (committee member), Nevins, Thomas A. (committee member), Schenck, Henry K. (committee member).*

Subjects/Keywords: Calabi-Yau threefold; Stable pairs; Deformation-obstruction theory; Derived categories; Equivariant cohomology; Virtual localization; Wallcrossing

…stability and we compute similar higher rank invariants using
the *wallcrossing* technique developed…

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APA (6^{th} Edition):

Sheshmani, A. (2011). Towards studying of the higher rank theory of stable pairs. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/26229

Chicago Manual of Style (16^{th} Edition):

Sheshmani, Artan. “Towards studying of the higher rank theory of stable pairs.” 2011. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed July 10, 2020. http://hdl.handle.net/2142/26229.

MLA Handbook (7^{th} Edition):

Sheshmani, Artan. “Towards studying of the higher rank theory of stable pairs.” 2011. Web. 10 Jul 2020.

Vancouver:

Sheshmani A. Towards studying of the higher rank theory of stable pairs. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2011. [cited 2020 Jul 10]. Available from: http://hdl.handle.net/2142/26229.

Council of Science Editors:

Sheshmani A. Towards studying of the higher rank theory of stable pairs. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2011. Available from: http://hdl.handle.net/2142/26229