Geodesic
On construction of positive closed currents with prescribed Lelong numbers
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 13 (2020) no. 3, pp. 331-341.

Voir la notice de l'article provenant de la source Math-Net.Ru

We establish that a sequence $(X_k)_{k\in\mathbb{N}}$ of analytic subsets of a domain $\Omega$ in $\mathbb{C}^n$, purely dimensioned, can be released as the family of upper-level sets for the Lelong numbers of some positive closed current. This holds whenever the sequence $(X_k)_{k\in\mathbb{N}}$ satisfies, for any compact subset $L$ of $\Omega$, the growth condition $\sum\limits_{k\in\mathbb{N}}C_k \hbox{mes}(X_k\cap L)\infty$. More precisely, we built a positive closed current $\Theta$ of bidimension $(p,p)$ on $\Omega$, such that the generic Lelong number $m_{X_k}$ of $\Theta$ along each $X_k$ satisfies $m_{X_k}=C_k$. In particular, we prove the existence of a plurisubharmonic function $v$ on $\Omega$ such that, each $X_k$ is contained in the upper-level set $E_{C_k}(dd^cv)$.
Keywords: closed positive current, plurisubharmonic function, potential, analytic set, Lelong number. 
@article{JSFU_2020_13_3_a6,
     author = {Hedi Khedhiri},
     title = {On construction of positive closed currents with prescribed {Lelong} numbers},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {331--341},
     publisher = {mathdoc},
     volume = {13},
     number = {3},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2020_13_3_a6/}
}
TY  - JOUR
AU  - Hedi Khedhiri
TI  - On construction of positive closed currents with prescribed Lelong numbers
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2020
SP  - 331
EP  - 341
VL  - 13
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2020_13_3_a6/
LA  - en
ID  - JSFU_2020_13_3_a6
ER  - 
%0 Journal Article
%A Hedi Khedhiri
%T On construction of positive closed currents with prescribed Lelong numbers
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2020
%P 331-341
%V 13
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2020_13_3_a6/
%G en
%F JSFU_2020_13_3_a6
Hedi Khedhiri. On construction of positive closed currents with prescribed Lelong numbers. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 13 (2020) no. 3, pp. 331-341. http://geodesic.mathdoc.fr/item/JSFU_2020_13_3_a6/

[1] H. Ben Messaoud, “Intermediate currents associated with a closed positive current”, Séminaire d'Analyse P. Lelong – P. Dolbeault – H. Skoda, Lecture Notes in Mathematics, 1028, 1983, 41–68 | DOI | MR

[2] H. Skoda, “New methods for the study of potentials associated with analytical sets”, Séminaire P. Lelong, Lectures Notes in Math., 410, 1972, 117–141 | DOI

[3] P.Lelong, “Sur la structure des courants positifs fermés”, Séminaire Pierre Lelong, Lecture Notes in Math., 578, Springer, Berlin, 1977, 136–156 | DOI | MR

[4] J.-P. Demailly, Complex Analytic and Differential Geometry, , 2012 http://www-fourier.ujf.-grenoble.fr/demailly/books.html

[5] Y.-T. Siu, “Analyticity of sets associated to Lelong numbers and the extension of closed positive currents”, Invent. Math., 27 (1974), 53–156 | DOI | MR | Zbl

[6] H. Skoda, “Sous ensembles analytiques d'ordre fini ou infini dans $\mathbb{C}^n$”, Bull. Soc. Math. France, 100 (1972), 353–408 | DOI | MR | Zbl