On construction of positive closed currents with prescribed Lelong numbers

Hedi Khedhiri

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 13 (2020) no. 3, pp. 331-341

Résumé

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 $E_{C_k}(dd^cv)$.

Détail

Keywords: closed positive current, plurisubharmonic function, potential, analytic set, Lelong number.