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 $\C^k$ 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 $\Cc^2$-smooth open subset of $V.$ We impose a mild reasonable condition on $T$ and $B,$ namely, $T$ is weakly approximable by $T_n^+-T_n^-$ on a neighborhood $U$ of $\overline B$ in $X,$ where $(T^\pm_n)_{n=1}^\infty$ are some positive plurisubharmonic $\Cc^3$-smooth forms of bidegree $(p,p)$ defined on $U$ such that the masses $\|T^\pm_n\|$ on $U$ are uniformly bounded and that the $\Cc^3$-norms of $T^\pm_n$ are uniformly bounded near $\partial B$ if $\partial B\not=\varnothing.$ Note that if $X$ is K\"ahler and $T$ is of class $\Cc^3$ near $\partial B,$ then the above mild condition is satisfied. In particular, this $\Cc^3$-smoothness near $\partial B$ is automatically fulfilled if either $\partial B=\varnothing$ or $V\cap \supp( T)\subset B.$ 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. We also show that $T$ admits tangent currents and that all tangent currents are not only positive plurisubharmonic, but also partially $V$-conic and partially pluriharmonic. When the currents $T^\pm_n$ are moreover pluriharmonic (resp. closed), we show, under a less restrictive smoothness of $T^\pm_n$ near $\partial B,$ that every tangent current is also $V$-conic pluriharmonic (resp. $V$-conic closed). We also prove that the generalized Lelong numbers are intrinsic. In fact, if we are only interested in the top degree Lelong number of $T$ along $B$ (that is, the $j$-th Lelong number for the maximal value $j=\min(l,k-p)$), then under a suitable holomorphic context, the above condition on the uniform regularity of $T^\pm_n$ near $\partial B$ can be removed. 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 strongly 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.