Let $X$ be a complex manifold of dimension $k,$ and $(V,ω)$ be a Kähler submanifold of dimension $l$ in $X,$ and $B\Subset V$ be a domain with $\mathcal{C}^2$-smooth boundary. Let $T$ be a positive plurisubharmonic current on $X$ such that $T$ satisfies a reasonable approximation condition on $X$ and near $\partial B.$ In our previous work we introduce the concept of the generalized Lelong numbers $ν_j(T,B)\in\mathbb{R}$ of $T$ along $B$ for $0\leq j\leq l.$ When $l=0,$ $V=B$ is a single point $x,$ $ν_0(T,B)$ is none other than the classical Lelong number of $T$ at $x.$
This article has five purposes: Firstly, we formulate the notion of the generalized Lelong number of $T$ associated to every closed smooth $(j,j)$-form on $V.$ This concept extends the previous notion of the generalized Lelong numbers. We also establish their basic properties. Secondly, we define the horizontal dimension $\hbar$ of such a current $T$ along $B.$ Next, we characterize $\hbar$ in terms of the generalized Lelong numbers. We also establish a Siu's upper-semicontinuity type theorem for the generalized Lelong numbers. In their above-mentioned context, Dinh and Sibony introduced some cohomology classes which may be regarded as their analogues of the classical Lelong numbers. Our third objective is to generalize their notion to the broader context where $T$ is (merely) positive pluriharmonic. Moreover, we also establish a formula relating Dinh-Sibony classes and the generalized Lelong numbers. Fourthly, we obtain an effective sufficient condition for defining the intersection of $m$ positive closed currents in the sense of Dinh-Sibony's theory of tangent currents on a compact Kähler manifold. Finally, we establish an effective sufficient condition for the continuity of the above intersection.
Lelong-Skoda transform for compact Kaehler manifolds and self-intersection inequalities
设$X$是维数为$k$的紧致Kaehler流形，$T$是$X$上维数为二维$(p,p)$（$1\leq p < k-1$）的正闭流.本文构造了二维$(k-1,k-1)$的$X$上的正闭合流$T$的连续线性变换$\mathcal{L}_p(T)$，它与$T$具有相同的Lelong数.由此导出了任意二次正闭流的自相交不等式。
