The generalized Lelong numbers and intersection theory

TL;DR

This paper extends the concept of Lelong numbers to generalized forms on complex manifolds, explores their properties, and applies these ideas to intersection theory of positive currents, including conditions for tangent current intersections.

Contribution

It introduces a broader notion of generalized Lelong numbers associated with smooth forms, characterizes their properties, and applies these to intersection theory of positive currents on Kähler manifolds.

Findings

Defined generalized Lelong numbers for smooth forms.

Established properties and semicontinuity of these numbers.

Provided conditions for intersection of positive currents.

Abstract

Let $X$ be a complex manifold of dimension $k,$ and $(V,\omega)$ be a K\"ahler 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 $\nu_j(T,B)\in\mathbb{R}$ of $T$ along $B$ for $0\leq j\leq l.$ When $l=0,$ $V=B$ is a single point $x,$ $\nu_0(T,B)$ 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 K\"ahler manifold. Finally, we establish an effective sufficient condition for the continuity of the above intersection.