English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Dinh--Sibony theory of tangent and density currents is a recent but powerfultool to study positive closed currents. Over twenty years ago, Alessandrini andBassanelli initiated the theory of the Lelong number of a positiveplurisubharmonic current in $\mathbb{C}^k$ along a linear subspace. Althoughthe latter theory is intriguing, it has not yet been explored in-depth sincethen. Introducing the concept of the generalized Lelong numbers and studyingthese new numerical values, we extend both theories to a more general class ofpositive plurisubharmonic currents and in a more general context of ambientmanifolds.  More specifically, in the first part of our article, we consider a positiveplurisubharmonic current $T$ of bidegree $(p,p)$ on a complex manifold $X$ ofdimension $k,$ 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 $B$ forevery $j$ with $\max(0,l-p)\leq j\leq \min(l,k-p)$ and prove their existence aswell 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 ofthe generalized Lelong numbers. As a consequence of this study, we show thatthe top degree Lelong number of $T$ along $B$ is totally intrinsic. This is ageneralization of the fundamental result of Siu (for positive closed currents)and of Alessandrini--Bassanelli (for positive plurisubharmonic currents) on theindependence of Lelong numbers at a single point on the choice of coordinates.

Positive plurisubharmonic currents: Generalized Lelong numbers and Tangent theorems