{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/113112"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/113112","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Toric mirror symmetry and GIT windows","abstract":"\"This thesis, partially based on the author's joint work [HZ20], studies mirror symmetry for toric varieties from the perspective of geometric invariant theory. We translate GIT constructions of toric varieties into skeletal terms and study them in the context of wrapped Fukaya categories and wrapped constructible sheaves. We explain how the mirror image of the Halpern-Leistner-Sam \"\"magic window\"\" for quasi-symmetric torus actions computes the category associated to the equivariant semistable skeleton upon stop removal, where the generating cotangent fibers are described by a generic shift of a lattice zonotope in the real character space of the torus. After identifying the \"\"FI parameter space\"\" of the B-model GLSM with the GKZ complex parameter space of the skeletal LG A-model, homological mirror symmetry in the quasi-symmetric case becomes two geometric incarnations of the Špenko-Van den Bergh schober over C^k stratified by a complexified periodic hyperplane arrangement whose complement models both parameter spaces. We obtain a universal deformation of skeletons interpolating variation of parameters.\"","abstract_html":""This thesis, partially based on the author's joint work [HZ20], studies mirror symmetry for toric varieties from the perspective of geometric invariant theory. We translate GIT constructions of toric varieties into skeletal terms and study them in the context of wrapped Fukaya categories and wrapped constructible sheaves. We explain how the mirror image of the Halpern-Leistner-Sam ""magic window"" for quasi-symmetric torus actions computes the category associated to the equivariant semistable skeleton upon stop removal, where the generating cotangent fibers are described by a generic shift of a lattice zonotope in the real character space of the torus. After identifying the ""FI parameter space"" of the B-model GLSM with the GKZ complex parameter space of the skeletal LG A-model, homological mirror symmetry in the quasi-symmetric case becomes two geometric incarnations of the Špenko-Van den Bergh schober over C^k stratified by a complexified periodic hyperplane arrangement whose complement models both parameter spaces. We obtain a universal deformation of skeletons interpolating variation of parameters."","abstract_has_math":false,"creators":["Huang, Jesse"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Pascaleff, James","Katz, Sheldon","Tolman, Susan","Dodd, Christopher"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-01-12T22:34:38Z","date_published":"2022-01-12T22:34:38Z","updated_at":"2026-07-22T22:24:53Z","subjects":["toric variety","mirror symmetry","window subcategory"],"languages":["en"],"rights":["Copyright 2021 Jesse Huang"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/113112","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Pascaleff, James","Katz, Sheldon","Tolman, Susan","Dodd, Christopher"]},{"key":"dc:creator","label":"Author","values":["Huang, Jesse"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2022-01-12T22:34:38Z","2024-01-12T22:35:30Z","2021-06-15","2021-08"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"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":["toric variety","mirror symmetry","window subcategory"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2021 Jesse Huang"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/113112"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"This thesis, partially based on the author's joint work [HZ20], studies mirror symmetry for toric varieties from the perspective of geometric invariant theory. We translate GIT constructions of toric varieties into skeletal terms and study them in the context of wrapped Fukaya categories and wrapped constructible sheaves. We explain how the mirror image of the Halpern-Leistner-Sam \"\"magic window\"\" for quasi-symmetric torus actions computes the category associated to the equivariant semistable skeleton upon stop removal, where the generating cotangent fibers are described by a generic shift of a lattice zonotope in the real character space of the torus. After identifying the \"\"FI parameter space\"\" of the B-model GLSM with the GKZ complex parameter space of the skeletal LG A-model, homological mirror symmetry in the quasi-symmetric case becomes two geometric incarnations of the Špenko-Van den Bergh schober over C^k stratified by a complexified periodic hyperplane arrangement whose complement models both parameter spaces. We obtain a universal deformation of skeletons interpolating variation of parameters.\"","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2023-08-01","The student, Jesse Huang, accepted the attached license on 2021-06-06 at 18:34.","The student, Jesse Huang, submitted this Dissertation for approval on 2021-06-07 at 11:46.","This Dissertation was approved for publication on 2021-06-15 at 10:46.","DSpace SAF Submission Ingestion Package generated from Vireo submission #16639 on 2022-01-12 at 12:51:22","Made available in DSpace on 2022-01-12T22:34:38Z (GMT). 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We translate GIT constructions of toric varieties into skeletal terms and study them in the context of wrapped Fukaya categories and wrapped constructible sheaves. We explain how the mirror image of the Halpern-Leistner-Sam \"\"magic window\"\" for quasi-symmetric torus actions computes the category associated to the equivariant semistable skeleton upon stop removal, where the generating cotangent fibers are described by a generic shift of a lattice zonotope in the real character space of the torus. After identifying the \"\"FI parameter space\"\" of the B-model GLSM with the GKZ complex parameter space of the skeletal LG A-model, homological mirror symmetry in the quasi-symmetric case becomes two geometric incarnations of the Špenko-Van den Bergh schober over C^k stratified by a complexified periodic hyperplane arrangement whose complement models both parameter spaces. We obtain a universal deformation of skeletons interpolating variation of parameters.\"","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2023-08-01","The student, Jesse Huang, accepted the attached license on 2021-06-06 at 18:34.","The student, Jesse Huang, submitted this Dissertation for approval on 2021-06-07 at 11:46.","This Dissertation was approved for publication on 2021-06-15 at 10:46.","DSpace SAF Submission Ingestion Package generated from Vireo submission #16639 on 2022-01-12 at 12:51:22","Made available in DSpace on 2022-01-12T22:34:38Z (GMT). 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