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Mathematics > Algebraic Geometry

arXiv:1805.11160 (math)
[Submitted on 28 May 2018]

Title:A non-Archimedean approach to K-stability

Authors:Sébastien Boucksom, Mattias Jonsson
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Abstract:We study K-stability properties of a smooth Fano variety X using non-Archimedean geometry, specifically the Berkovich analytification of X with respect to the trivial absolute value on the ground field. More precisely, we view K-semistability and uniform K-stability as conditions on the space of plurisubharmonic (psh) metrics on the anticanonical bundle of X. Using the non-Archimedean Calabi-Yau theorem and the Legendre transform, this allows us to give a new proof that K-stability is equivalent to Ding stability. By choosing suitable psh metrics, we also recover the valuative criterion of K-stability by Fujita and Li. Finally, we study the asymptotic Fubini-Study operator, which associates a psh metric to any graded filtration (or norm) on the anticanonical ring. Our results hold for arbitrary smooth polarized varieties, and suitable adjoint/twisted notions of K-stability and Ding stability. They do not rely on the Minimal Model Program.
Subjects: Algebraic Geometry (math.AG); Complex Variables (math.CV)
MSC classes: 32Q26, 32P05, 32U05, 14J45, 14G22
Cite as: arXiv:1805.11160 [math.AG]
  (or arXiv:1805.11160v1 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.1805.11160

Submission history

From: Mattias Jonsson [view email]
[v1] Mon, 28 May 2018 20:20:55 UTC (48 KB)
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