{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/145043"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/145043","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"K-stability of Log Fano Cone Singularities","abstract":"In this thesis, we define the 𝛿-invariant for log Fano cone singularities, and show that the necessary and sufficient condition for K-semistability is 𝛿 ≥ 1. This generalizes the result of C. Li and K. Fujita. We also prove that on any log Fano cone singularity of dimension 𝑛 whose 𝛿-invariant is less than (𝑛+1)/𝑛, any valuation computing 𝛿 has a finitely generated associated graded ring. This shows a log Fano cone is K-polystable if and only if it is uniformly K-stable. Together with earlier works, this implies the Yau-Tian-Donaldson Conjecture for Fano cone.","abstract_html":"In this thesis, we define the 𝛿-invariant for log Fano cone singularities, and show that the necessary and sufficient condition for K-semistability is 𝛿 ≥ 1. This generalizes the result of C. Li and K. Fujita. We also prove that on any log Fano cone singularity of dimension 𝑛 whose 𝛿-invariant is less than (𝑛+1)/𝑛, any valuation computing 𝛿 has a finitely generated associated graded ring. This shows a log Fano cone is K-polystable if and only if it is uniformly K-stable. Together with earlier works, this implies the Yau-Tian-Donaldson Conjecture for Fano cone.","abstract_has_math":false,"creators":["Huang, Kai"],"institution":"Massachusetts Institute of Technology","degree_name":"Doctoral","degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. 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This generalizes the result of C. Li and K. Fujita. We also prove that on any log Fano cone singularity of dimension 𝑛 whose 𝛿-invariant is less than (𝑛+1)/𝑛, any valuation computing 𝛿 has a finitely generated associated graded ring. This shows a log Fano cone is K-polystable if and only if it is uniformly K-stable. Together with earlier works, this implies the Yau-Tian-Donaldson Conjecture for Fano cone."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["K-stability of Log Fano Cone Singularities"]}]}],"canonical_facts":{"dc:contributor.advisor":["Xu, Chenyang"],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Mathematics"],"dc:creator":["Huang, Kai"],"dc:date.accessioned":["2022-08-29T16:29:10Z"],"dc:date.available":["2022-08-29T16:29:10Z"],"dc:date.issued":["2022-05"],"dc:description.abstract":["In this thesis, we define the 𝛿-invariant for log Fano cone singularities, and show that the necessary and sufficient condition for K-semistability is 𝛿 ≥ 1. This generalizes the result of C. Li and K. Fujita. We also prove that on any log Fano cone singularity of dimension 𝑛 whose 𝛿-invariant is less than (𝑛+1)/𝑛, any valuation computing 𝛿 has a finitely generated associated graded ring. This shows a log Fano cone is K-polystable if and only if it is uniformly K-stable. Together with earlier works, this implies the Yau-Tian-Donaldson Conjecture for Fano cone."],"dc:description.degree":["Ph.D."],"dc:identifier.uri":["https://hdl.handle.net/1721.1/145043"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["In Copyright - Educational Use Permitted","Copyright MIT"],"dc:rights.uri":["http://rightsstatements.org/page/InC-EDU/1.0/"],"dc:title":["K-stability of Log Fano Cone Singularities"],"dc:type":["Thesis"],"thesis:degree_name":["Doctoral","Doctor of Philosophy"]},"updated_at":"2026-07-22T22:22:13Z"}