Weight functions on non-Archimedean analytic spaces and the Kontsevich–Soibelman skeleton

Abstract

We associate a weight function to pairs (X, ω) consisting of a smooth and proper variety X over a complete discretely valued field and a pluricanonical form ω on X. This weight function is a real-valued function on the non-Archimedean analytification of X. It is piecewise affine on the skeleton of any regular model with strict normal crossings of X, and strictly ascending as one moves away from the skeleton. We apply these properties to the study of the Kontsevich-Soibelman skeleton of (X, ω), and we prove that this skeleton is connected when X has geometric genus one and ω is a canonical form on X. This result can be viewed as an analog of the Shokurov-Kollár connectedness theorem in birational geometry.

Convergence of p-adic pluricanonical measures to Lebesgue measures on skeleta in Berkovich spaces

“…The Kontsevich-Soibelman skeleton. Let X be a connected smooth and proper K-scheme, and let θ be a non-zero m-canonical form on X, for some positive integer m. It is explained in [MN15] how one can attach to θ a canonical subspace of X an , called the Kontsevich-Soibelman skeleton of the pair (X, θ). Such an object first appeared in the work of Kontsevich and Soibelman on the non-archimedean SYZ fibration [KS06].…”

“…Remark 4.4.2. In [MN15], the definition of the weight function was extended to all monomial points in X an (and then further to X an by means of an approximation procedure, assuming resolution of singularities). The formula in Proposition 4.4.1 is valid for all the points in Sk(X ), by the same proof.…”

“…First, the measure λ a is supported on the top-dimensional part of the Kontsevich-Soibelman skeleton Sk(X, θ) ⊂ X an of (X, θ) [KS06,MN15], defined as the closure in X an of the locus where a certain function, the weight function wt θ attains its infimum, see Section 2.2. If X has a log smooth (or merely log regular) model, then Sk(X, θ) is nonempty and carries a natural piecewise integral affine structure, and thus a natural Lebesgue measure (see Section 5.1); the measure λ a equals the Lebesgue measure on the top dimensional part of Sk(X, θ), suitably weighted to make it well-behaved under tame finite extensions of the base field K; we call it the stable Lebesgue measure.…”

Tropical dynamics of area-preserving maps

“…In general, to a point p ∈ ∆(X ) there is associated a monomial valuation and monomial point as follows (see [MN15,Prop. 2…”

Chern–Weil Theory for Line Bundles with the Family Arakelov Metric

“…From Section 4 on we focus on the particular case of surfaces fibered over a curve. Using the notion of skeleton from [3] [27] we give a presentation of the group of b-divisors from a metric and functional perspective, leading to the isomorphism (1.7). We also briefly touch upon the connection with the theory of Berkovich analytic curves.…”