Khalil Zahmoul | Searcharxiv

arXiv2021

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

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.

math.CV↗

arXiv2019

Lelong-Jensen formula, Demailly-Lelong numbers and weighted degree of positive supercurrents

The goal of this work is to extend the concepts of generalized Lelong number of positive currents investigated by Skoda, Demailly and Ghiloufi in complex analysis, to weakly positive supercurrents on the real superspaces. We generalize then a result of Lagerberg when the supercurrent is closed as well as a very recent result of Berndtsson for minimal supercurrents associated to submanifolds of $\mathbb{R}^n$. The main tool is a variant of the well-known Lelong-Jensen formula in the superformalism case. Moreover, we extend to our setting various interesting theorems in complex analysis such as Demailly and Rashkovskii comparison theorems. We also complete the work begun by Lagerberg on the degree of positive closed supercurrents and we prove a removable singularities result for positive supercurrents.

math.CV↗