Poincar\'e-Lelong formula, $J$-analytic subsets and Lelong numbers of currents on almost complex manifolds

TL;DR

This paper extends classical complex analysis tools to almost complex manifolds by establishing a Poincaré-Lelong formula, defining J-analytic subsets, and proving the invariance of Lelong numbers for currents.

Contribution

It introduces a Poincaré-Lelong formula in the almost complex setting, defines J-analytic subsets, and proves Lelong number invariance, advancing the understanding of currents in almost complex geometry.

Findings

Poincaré-Lelong formula established for almost complex manifolds

J-analytic subsets introduced and studied

Lelong numbers are coordinate-independent

Abstract

In this paper, we first establish a Poincar\'e-Lelong type formula in the almost complex setting. Then, after introducing the notion of $J$-analytic subsets, we study the restriction of a closed positive current defined in an almost complex manifold $(M,J)$ on a $J$-analytic subset. Finally, we prove that the Lelong numbers of a plurisubharmonic current defined on an almost complex manifold are independent of the coordinate systems.