{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/358793"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/358793","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Mirrors to Toric Degenerations via Intrinsic Mirror Symmetry","abstract":"We explore the connection between two mirror constructions in Gross-Siebert mirror symmetry: toric degeneration mirror symmetry and intrinsic mirror symmetry. After briefly exploring the case of degenerations of elliptic curves, we show that the Gross-Siebert mirror construction for minimal relative log Calabi-Yau degenerations generalizes that for divisorial toric degenerations $\\bar{\\mathfrak{X}} \\to \\mathcal{S}$ of K3-s that have a smooth generic fibre. We achieve this by constructing a resolution of $\\bar{\\mathfrak{X}} \\to \\mathcal{S}$ to a relative minimal log Calabi-Yau degeneration $\\mathfrak{X} \\to \\mathcal{S}$ and comparing the algorithmic scattering diagram $\\bar{\\mathfrak{D}}$ giving rise to the toric degeneration mirror $\\check{\\bar{\\mathfrak{X}}}$ and the canonical scattering diagram $\\mathfrak{D}$ giving rise to the intrinsic mirror $\\check{\\mathfrak{X}}$. Moreover, we vastly expand the construction and obtain a correspondence between the restriction of the intrinsic mirror to the (numerical) minimal relative Gross-Siebert locus and the universal toric degeneration mirror. We also discuss generalizing the results to higher dimensions. In particular, we construct log smooth resolutions for a natural family of toric degenerations of Calabi-Yau threefolds.","abstract_html":"We explore the connection between two mirror constructions in Gross-Siebert mirror symmetry: toric degeneration mirror symmetry and intrinsic mirror symmetry. After briefly exploring the case of degenerations of elliptic curves, we show that the Gross-Siebert mirror construction for minimal relative log Calabi-Yau degenerations generalizes that for divisorial toric degenerations $\\bar{\\mathfrak{X}} \\to \\mathcal{S}$ of K3-s that have a smooth generic fibre. We achieve this by constructing a resolution of $\\bar{\\mathfrak{X}} \\to \\mathcal{S}$ to a relative minimal log Calabi-Yau degeneration $\\mathfrak{X} \\to \\mathcal{S}$ and comparing the algorithmic scattering diagram $\\bar{\\mathfrak{D}}$ giving rise to the toric degeneration mirror $\\check{\\bar{\\mathfrak{X}}}$ and the canonical scattering diagram $\\mathfrak{D}$ giving rise to the intrinsic mirror $\\check{\\mathfrak{X}}$. Moreover, we vastly expand the construction and obtain a correspondence between the restriction of the intrinsic mirror to the (numerical) minimal relative Gross-Siebert locus and the universal toric degeneration mirror. We also discuss generalizing the results to higher dimensions. In particular, we construct log smooth resolutions for a natural family of toric degenerations of Calabi-Yau threefolds.","abstract_has_math":true,"creators":["Goncharov, Evgeny"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Gross, Mark"],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-06-01","date_published":"2023-06-01","updated_at":"2026-07-22T22:23:54Z","subjects":["Logarithmic geometry","Mirror symmetry","Resolution of singularities","Toric degenerations"],"languages":["eng"],"rights":[],"rights_urls":["https://www.repository.cam.ac.uk/bitstreams/e0bb9e61-8407-4cb4-a4b1-d81419e7ab99/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.102243","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Gross, Mark"]},{"key":"dc:creator","label":"Author","values":["Goncharov, Evgeny"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2023-06-01"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/358793"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Logarithmic geometry","Mirror symmetry","Resolution of singularities","Toric degenerations"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["https://www.repository.cam.ac.uk/bitstreams/e0bb9e61-8407-4cb4-a4b1-d81419e7ab99/download","https://www.rioxx.net/licenses/all-rights-reserved/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["https://doi.org/10.17863/CAM.102243"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://www.repository.cam.ac.uk/bitstreams/c08d05fd-48db-415b-b32f-6a823ad30bb5/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We explore the connection between two mirror constructions in Gross-Siebert mirror symmetry: toric degeneration mirror symmetry and intrinsic mirror symmetry. After briefly exploring the case of degenerations of elliptic curves, we show that the Gross-Siebert mirror construction for minimal relative log Calabi-Yau degenerations generalizes that for divisorial toric degenerations $\\bar{\\mathfrak{X}} \\to \\mathcal{S}$ of K3-s that have a smooth generic fibre. We achieve this by constructing a resolution of $\\bar{\\mathfrak{X}} \\to \\mathcal{S}$ to a relative minimal log Calabi-Yau degeneration $\\mathfrak{X} \\to \\mathcal{S}$ and comparing the algorithmic scattering diagram $\\bar{\\mathfrak{D}}$ giving rise to the toric degeneration mirror $\\check{\\bar{\\mathfrak{X}}}$ and the canonical scattering diagram $\\mathfrak{D}$ giving rise to the intrinsic mirror $\\check{\\mathfrak{X}}$. Moreover, we vastly expand the construction and obtain a correspondence between the restriction of the intrinsic mirror to the (numerical) minimal relative Gross-Siebert locus and the universal toric degeneration mirror. We also discuss generalizing the results to higher dimensions. In particular, we construct log smooth resolutions for a natural family of toric degenerations of Calabi-Yau threefolds."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["8cb8a3f696de34ddf6744f001507a020","87eda9de84448d1f82354d60eee3eb5f"]},{"key":"dc:title","label":"Title","values":["Mirrors to Toric Degenerations via Intrinsic Mirror Symmetry"]}]}],"canonical_facts":{"dc:contributor.advisor":["Gross, Mark"],"dc:creator":["Goncharov, Evgeny"],"dc:date.issued":["2023-06-01"],"dc:description.abstract":["We explore the connection between two mirror constructions in Gross-Siebert mirror symmetry: toric degeneration mirror symmetry and intrinsic mirror symmetry. After briefly exploring the case of degenerations of elliptic curves, we show that the Gross-Siebert mirror construction for minimal relative log Calabi-Yau degenerations generalizes that for divisorial toric degenerations $\\bar{\\mathfrak{X}} \\to \\mathcal{S}$ of K3-s that have a smooth generic fibre. We achieve this by constructing a resolution of $\\bar{\\mathfrak{X}} \\to \\mathcal{S}$ to a relative minimal log Calabi-Yau degeneration $\\mathfrak{X} \\to \\mathcal{S}$ and comparing the algorithmic scattering diagram $\\bar{\\mathfrak{D}}$ giving rise to the toric degeneration mirror $\\check{\\bar{\\mathfrak{X}}}$ and the canonical scattering diagram $\\mathfrak{D}$ giving rise to the intrinsic mirror $\\check{\\mathfrak{X}}$. Moreover, we vastly expand the construction and obtain a correspondence between the restriction of the intrinsic mirror to the (numerical) minimal relative Gross-Siebert locus and the universal toric degeneration mirror. We also discuss generalizing the results to higher dimensions. In particular, we construct log smooth resolutions for a natural family of toric degenerations of Calabi-Yau threefolds."],"dc:format.checksum.md5":["8cb8a3f696de34ddf6744f001507a020","87eda9de84448d1f82354d60eee3eb5f"],"dc:identifier.doi":["https://doi.org/10.17863/CAM.102243"],"dc:identifier.uri":["https://www.repository.cam.ac.uk/bitstreams/c08d05fd-48db-415b-b32f-6a823ad30bb5/download"],"dc:language":["eng"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/358793"],"dc:rights":["https://www.repository.cam.ac.uk/bitstreams/e0bb9e61-8407-4cb4-a4b1-d81419e7ab99/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"dc:subject":["Logarithmic geometry","Mirror symmetry","Resolution of singularities","Toric degenerations"],"dc:title":["Mirrors to Toric Degenerations via Intrinsic Mirror Symmetry"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-22T22:23:54Z"}