Positive plurisubharmonic currents: Generalized Lelong numbers and Tangent theorems

TL;DR

This paper extends the theory of Lelong numbers for positive plurisubharmonic currents to more general contexts, introducing generalized Lelong numbers and tangent theorems, with applications to complex manifolds and intrinsic geometric properties.

Contribution

It introduces generalized Lelong numbers for positive plurisubharmonic currents on complex manifolds and establishes their fundamental properties and geometric characterizations.

Findings

Defined the j-th Lelong number along submanifolds

Proved existence and basic properties of these Lelong numbers

Showed the top degree Lelong number is intrinsic and coordinate-independent

Abstract

Dinh--Sibony theory of tangent and density currents is a recent but powerful tool to study positive closed currents. Over twenty years ago, Alessandrini and Bassanelli initiated the theory of the Lelong number of a positive plurisubharmonic current in $\mathbb{C}^k$ along a linear subspace. Although the latter theory is intriguing, it has not yet been explored in-depth since then. Introducing the concept of the generalized Lelong numbers and studying these new numerical values, we extend both theories to a more general class of positive plurisubharmonic currents and in a more general context of ambient manifolds. More specifically, in the first part of our article, we consider a positive plurisubharmonic current $T$ of bidegree $(p,p)$ on a complex manifold $X$ of dimension $k,$ and let $V\subset X$ be a K\"ahler submanifold of dimension $l$ and $B$ a relatively compact piecewise $\mathcal{C}^2$-smooth open subset of $V.$ We define the notion of the $j$-th Lelong number of $T$ along $B$ for every $j$ with $\max(0,l-p)\leq j\leq \min(l,k-p)$ and prove their existence as well as their basic properties. Our method relies on some Lelong-Jensen formulas for the normal bundle to $V$ in $X,$ which are of independent interest. The second part of our article is devoted to geometric characterizations of the generalized Lelong numbers. As a consequence of this study, we show that the top degree Lelong number of $T$ along $B$ is totally intrinsic. This is a generalization of the fundamental result of Siu (for positive closed currents) and of Alessandrini--Bassanelli (for positive plurisubharmonic currents) on the independence of Lelong numbers at a single point on the choice of coordinates.