2026-09-23T17:28:03Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/26229 2023-07-10 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Katz, Sheldon Nevins, Thomas A. Bradlow, Steven B. Katz, Sheldon Nevins, Thomas A. Schenck, Henry K. Sheshmani, Artan 2011-08-25T22:19:38Z 2011-08-25T22:19:38Z 2011-08-25T22:19:38Z 2011-08 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. Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2011-07-04T15:41:00Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 4 Sheshmani_Artan.pdf: 972533 bytes, checksum: a9eef18958230813159cec66af7bf7fa (MD5) Thesis-final-draft-2011-0620.tex: 997007 bytes, checksum: 8b5dc10299e73cabd6406a365556b9dd (MD5) Thesis-final-draft-2011-0620.bbl: 5922 bytes, checksum: 21bd3ba1123e328b6668b88d5d733bfc (MD5) Sheshmani_Artan.pdf: 972512 bytes, checksum: 5a81e9d753e49a36b443b4808da1d138 (MD5) Made available in DSpace on 2011-08-25T22:19:38Z (GMT). No. of bitstreams: 4 Sheshmani_Artan.pdf: 972389 bytes, checksum: bf55b83132b66670d5b2d393f76069fc (MD5) Thesis-final-draft-2011-0620.bbl: 5781 bytes, checksum: 2ca10c61b546f73b66de00ddedcf5659 (MD5) license.txt: 4065 bytes, checksum: 68c4edda23b105a8e668bb6a3c606a38 (MD5) Thesis-final-draft-2011-0620.tex: 997020 bytes, checksum: 8a06412b7aea7b594052a527f3b8b65a (MD5) http://hdl.handle.net/2142/26229 en Copyright 2011 Artan Sheshmani Calabi-Yau threefold Stable pairs Deformation-obstruction theory Derived categories Equivariant cohomology Virtual localization Wallcrossing Towards studying of the higher rank theory of stable pairs Mathematics 1257 Mathematics 0439 University of Illinois at Urbana-Champaign Dissertation Ph.D. PHD:Mathematics -UIUC 10KS0439PHD