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K-stability of Log Fano Cone Singularities

Author(s)
Huang, Kai
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Advisor
Xu, Chenyang
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In Copyright - Educational Use Permitted Copyright MIT http://rightsstatements.org/page/InC-EDU/1.0/
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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.
Date issued
2022-05
URI
https://hdl.handle.net/1721.1/145043
Department
Massachusetts Institute of Technology. Department of Mathematics
Publisher
Massachusetts Institute of Technology

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