{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/26229"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/26229","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Towards studying of the higher rank theory of stable pairs","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.","abstract_html":"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 <span class=\"etd-inline-math\">\\mathcal{O}<sub>X</sub><sup>\\oplus r</sup>(-n)\\xrightarrow{\\phi} F</span> 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 <span class=\"etd-inline-math\">\\mathbb{P}<sup>1</sup></span> 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.","abstract_has_math":true,"creators":["Sheshmani, Artan"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Katz, Sheldon","Nevins, Thomas A.","Bradlow, Steven B.","Schenck, Henry K."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-08-25T22:19:38Z","date_published":"2011-08-25T22:19:38Z","updated_at":"2026-07-22T22:25:26Z","subjects":["Calabi-Yau threefold","Stable pairs","Deformation-obstruction theory","Derived categories","Equivariant cohomology","Virtual localization","Wallcrossing"],"languages":["en"],"rights":["Copyright 2011 Artan Sheshmani"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/26229","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Katz, Sheldon","Nevins, Thomas A.","Bradlow, Steven B.","Schenck, Henry K."]},{"key":"dc:creator","label":"Author","values":["Sheshmani, Artan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-08-25T22:19:38Z","2011-08"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Calabi-Yau threefold","Stable pairs","Deformation-obstruction theory","Derived categories","Equivariant cohomology","Virtual localization","Wallcrossing"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2011 Artan Sheshmani"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/26229"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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). 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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). 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