On semipositive piecewise linear functions in non-archimedean analytic geometry

TL;DR

The paper generalizes Zhang's semipositive model metric framework to strictly analytic spaces over a nontrivially valued non-Archimedean field, establishing a robust theory for semipositive $\R$-PL functions on $k$-analytic spaces. Central to the approach are two lifting theorems (affine and local) that connect reductions of formal models to the analytic generic fiber, together with a refined affinoid Bieri–Groves-type result for tropicalizations. Key contributions include the equivalence between semipositivity and curve-restriction, stability under pointwise limits and maxima, and a maximum principle for piecewise smooth psh functions, all within a framework that blends non-archimedean pluripotential theory with tropical geometry. These results lay groundwork for non-archimedean height theory and tropical-analytic methods paralleling complex pluripotential theory, with potential applications to Zhang's heights and related arithmetic problems.

Abstract

In this paper, we generalize results on Zhang's semipositive model metrics from the algebraic setting to strictly analytic spaces over a non-trivially valued non-Archimedean field. We prove stability under pointwise limits and under forming the maximum. We also prove the maximum principle. As tools, we use a local lifting theorem from the special fiber to the generic fiber of a model and an affinoid Bieri--Groves theorem.

Theorems & Definitions (39)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Remark 7
• Definition 8
• Theorem 9: Affine Lifting Theorem
• Lemma 10