$m$-potential theory and $m$-generalized Lelong numbers associated with $m$-positive supercurrents

TL;DR

This paper extends potential theory and Lelong numbers to the setting of supercurrents, defining new potentials, studying the superHessian operator, and generalizing results from complex Hessian theory to superformalism.

Contribution

It introduces the local potential for weakly positive supercurrents, studies the superHessian operator, and generalizes Demailly-Lelong numbers within superformalism.

Findings

Defined local potentials for supercurrents.

Established continuity of the superHessian operator.

Generalized Demailly-Lelong numbers and introduced Cegrell-type classes.

Abstract

In this study, we first define the local potential associated to a weakly positive closed supercurrent in analogy to the one investigated by Ben Messaoud and El Mir in the complex setting. Next, we study the definition and the continuity of the $m$-superHessian operator for unbounded $m$-convex functions. As an application, we generalize our previous work on Demailly-Lelong numbers and several related results in the superformalism setting. Furthermore, strongly inspired by the complex Hessian theory, we introduce the Cegrell-type classes as well as a generalization of some $m$-potential results in the class of $m$-convex functions.