Jawhar Hbil | Searcharxiv

arXiv2014

Pluricomplex energy classes associated to a positive closed current

The aim of this paper is to extend the domain of definition of $(dd^c\centerdot)^q\wedge T$ on some classes of plurisubharmonic (psh) functions, which are not necessary bounded, where $T$ is a positive closed current of bidimension $(q,q)$ on an open set $Ω$ of $\Bbb C^n$. We introduce two classes $\mathcal{F}_{p}^{T}(Ω)$ and $\mathcal{E}_p^T(Ω)$ and we show that they belong to the domain of definition of the operator $(dd^c\centerdot)^q\wedge T$. We also prove that all functions belong to these classes are $C_T$-quasicontinuous and that the comparison principle is valid in them.

math.CV↗

arXiv2012

Existence of the directional tangent cone to a positive current

In this paper, we start by proving the existence of the strict transform of a positive current $T$ as soon as its $j^{th}$ currents, $T_j$, are plurisubharmonics or plurisuperharmonics. Then, with a suitable condition on $T_j$, we show the existence of the directional tangent cone to $T$. In the particular case, when $T$ is closed, we give a second condition independent to the previous one.

math.CV↗