Lelong–Jensen formula, Demailly–Lelong numbers and weighted degree of 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.

Keywords:

• Superforms
• supercurrents
• Hessian operator
• Lelong numbers

AMS SUBJECT CLASSIFICATIONS:

• 32U05
• 32U25
• 32U40
• 31C05

Acknowledgments

We would like to express our gratitude towards Professor Bo Berndtsson for his precious observations that helped to improve this article. Also, we would like to thank the referee for the remarks given, which helped enhance the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).