A new generalization of the Lelong number

Aron Lagerberg Department of Mathematics, Chalmers University of Technology and University of Göteborg

Complex Variables and Complex Analysis mathscidoc:1701.08009

Arkiv for Matematik, 51, (1), 125-156, 2010.8
We will introduce a quantity which measures the singularity of a plurisubharmonic function$φ$relative to another plurisubharmonic function$ψ$, at a point$a$. We denote this quantity by$ν$_{$a$,$ψ$}($φ$). It can be seen as a generalization of the classical Lelong number in a natural way: if$ψ$=($n$−1)log| ⋅ −$a$|, where$n$is the dimension of the set where$φ$is defined, then$ν$_{$a$,$ψ$}($φ$) coincides with the classical Lelong number of$φ$at the point$a$. The main theorem of this article says that the upper level sets of our generalized Lelong number, i.e. the sets of the form {$z$:$ν$_{$z$,$ψ$}($φ$)≥$c$} where$c$>0, are in fact analytic sets, provided that the$weight$$ψ$satisfies some additional conditions.
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  title={A new generalization of the Lelong number},
  author={Aron Lagerberg},
  booktitle={Arkiv for Matematik},
Aron Lagerberg. A new generalization of the Lelong number. 2010. Vol. 51. In Arkiv for Matematik. pp.125-156. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203634614441036.
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