Formes diff\'erentielles r\'eelles et courants sur les espaces de Berkovich

TL;DR

This paper develops a comprehensive theory of differential forms, currents, and intersection theory on Berkovich spaces, bridging tropical geometry and non-archimedean analysis with applications to algebraic and arithmetic geometry.

Contribution

It introduces a novel framework for differential forms and currents on Berkovich spaces, extending tropical and non-archimedean geometry tools to algebraic and arithmetic contexts.

Findings

Constructs canonical calibrations of skeleta for integration.

Establishes analogues of classical formulas like Poincaré-Lelong and Stokes.

Links intersection theory of line bundles with non-archimedean metrics.

Abstract

We define a theory of real $(p,q)$-forms and currents on Berkovich spaces which is parallel to the theory of differential forms on complex spaces. It is based on Lagerberg's theory of superforms in tropical geometry and on the consideration of tropicalization maps and skeleta on domains of non archimedean analytic spaces in the sense of Berkovich. We construct canonical calibrations of skeleta of analytic spaces, which give rise to integrals of $(n,n)$-forms, and a variant of Stokes formula. The theory of currents furnishes analogues of the Poincar\'e-Lelong formula, as well as the formulas of Bochner-Martinelli and Levine. We define a notion of plurisubharmonic functions and develop an analogue of Bedford-Taylor's theory of products of closed positive currents. Smooth metrized line bundles have a Chern form; the integrals of products of these Chern forms is compatible with numerical intersection theory. The case of psh metrics gives rise to Chern currents. In the case of formal metrics, we compute these product currents in terms of intersection numbers of the special fiber. In a final chapter, we detail how the uniformization of abelian varieties allows to study the canonical metrics on their line bundles. The theory allows to reinterpret tropical intersection theory and is presented in the general context of so-called "tropical spaces" which we introduce in a first part of the book.