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

TL;DR

This paper extends the concepts of Lelong numbers and related theorems from complex analysis to positive supercurrents on real superspaces, introducing new formulas and generalizations.

Contribution

It generalizes Lelong-Jensen formulas and key theorems to the setting of weakly positive supercurrents, advancing the theory in real superspaces.

Findings

Generalized Lelong-Jensen formula for supercurrents

Extended comparison theorems in the superformalism setting

Proved a removable singularities theorem for positive supercurrents

Abstract

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.