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

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.

This is the second version of the paper. We added new results and we improved the presentation of the article