m-Pluripotential Theory on Riemannian Spaces and Tropical Geometry

TL;DR

This paper extends m-pluripotential theory to Riemannian spaces, exploring positive supercurrents, tropical varieties, and generalizing classical results to new geometric and analytical contexts.

Contribution

It introduces a framework connecting m-pluripotential theory with Riemannian superspaces and tropical geometry, including new capacity notions and geometric characterizations.

Findings

Relation between positive supercurrents and tropical varieties clarified

Generalization of Cartan's quasicontinuity to Riemannian m-subharmonic functions

Introduction of indicators and Newton numbers for m-subharmonic functions

Abstract

In this study we extend the concepts of $m$-pluripotential theory to the Riemannian superspace formalism. Since in this setting positive supercurrents and tropical varieties are closely related, we try to understand the relative capacity notion with respect to the intersection of tropical hypersurfaces. Moreover, we generalize the classical quasicontinuity result of Cartan to $m$-subharmonic functions of Riemannian spaces and lastly we introduce the indicators of $m$-subharmonic functions and give a geometric characterization of their Newton numbers.