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.