Approches courantielles \`a la Mellin dans un cadre non archim\'edien

TL;DR

This paper introduces a Mellin-type approach for approximating Green currents and integration currents in non-Archimedean Berkovich spaces, extending classical formulas like Crofton and King to this setting.

Contribution

It develops a Mellin-based method for Green current approximation and adapts Crofton and King formulas to non-Archimedean Berkovich spaces, including Vogel and Segre currents.

Findings

Mellin approach effectively approximates Green currents in non-Archimedean spaces.

Extension of Crofton and King formulas to Berkovich spaces.

Approximate realization of Vogel and Segre currents in the non-Archimedean context.

Abstract

We propose an approach of Mellin type for the approximation of integration currents or the effective realization of normalized Green currents associated with a cycle $ \bigwedge_1^m[{\rm div} (s_j)] $, where $s_j $ is a meromorphic section of a line bundle $ \mathscr{L}_j \rightarrow U$ over an open $U$ in a good Berkovich space when each $ \mathscr{L}_j$ has a smooth metric and $ {\rm codim}_{U}\big (\bigcap_{j \in J} {\rm Supp} [{\rm div (s_j)}] \big)\geq \# J$ for every set $ J \subset \{1, ..., p \} $. We also study the transposition to the non archimedean context of Crofton and King formulas, particularly the approximate realization of Vogel and Segre currents.