Towards studying of the higher rank theory of stable pairs

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2011

Author dc:creator

• Sheshmani, Artan

Contributors dc:contributor

• Katz, Sheldon
• Nevins, Thomas A.
• Bradlow, Steven B.
• Schenck, Henry K.

• Calabi-Yau threefold
• Stable pairs
• Deformation-obstruction theory
• Derived categories
• Equivariant cohomology
• Virtual localization
• Wallcrossing

Statement dc:rights

• Copyright 2011 Artan Sheshmani

Language dc:language

en

Handle dc:identifier

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

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/26229