Positive plurisubharmonic currents: Generalized Lelong numbers and Tangent theorems

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.