A non-Archimedean approach to K-stability

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.

References in corpus (12)

• Tian's properness conjectures and Finsler geometry of the space of Kahler metrics
• A variational approach to the Yau-Tian-Donaldson conjecture
• Formes différentielles réelles et courants sur les espaces de Berkovich
• Seshadri constants, Diophantine approximation, and Roth's Theorem for arbitrary varieties
• Existence of valuations with smallest normalized volume
• Stability of Valuations and Kollár Components
• A tropical approach to non-archimedean Arakelov theory
• The normalized volume of a singularity is lower semicontinuous
• A valuative criterion for uniform K-stability of $\mathbb{Q}$-Fano varieties
• Kähler-Einstein metrics and volume minimization
• Differentiability of non-archimedean volumes and non-archimedean Monge-Ampère equations (with an appendix by Robert Lazarsfeld)
• Minimizing normalized volumes of valuations

Cited by in corpus (20)

• Basis log canonical thresholds, local intersection estimates, and asymptotically log del Pezzo surfaces
• Optimal destabilization of K-unstable Fano varieties via stability thresholds
• Geodesic rays and stability in the cscK problem
• Openness of uniform K-stability in families of $\mathbb{Q}$-Fano varieties
• K-stability of Fano varieties: an algebro-geometric approach
• The uniform version of Yau-Tian-Donaldson conjecture for singular Fano varieties
• Uniqueness of K-polystable degenerations of Fano varieties
• The existence of the Kähler-Ricci soliton degeneration
• Differentiability of relative volumes over an arbitrary non-Archimedean field
• The closures of test configurations and algebraic singularity types
• Continuity of delta invariants and twisted Kähler--Einstein metrics
• Geometric flow, Multiplier ideal sheaves and Optimal destabilizer for a Fano manifold
• Mabuchi geometry of big cohomology classes with prescribed singularities
• Openness of uniformly valuative stability on the Kähler cone of projective manifolds
• Volume and Monge-Ampère energy on polarized affine varieties
• Emergent complex geometry
• Entropies in $\mu$-framework of canonical metrics and K-stability, I -- Archimedean aspect: Perelman's W-entropy and $\mu$-cscK metrics
• Valuative invariants with higher moments
• Delta invariants of projective bundles and projective cones of Fano type
• The Kähler-Ricci flow and quantitative bounds for Donaldson-Futaki invariants of optimal degenerations