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arXiv · 1805.11160

A non-Archimedean approach to K-stability

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.

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Sébastien Boucksom, Mattias Jonsson. 2018-05-28. A non-Archimedean approach to K-stability. https://arxiv.org/abs/1805.11160

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