Existence of the directional tangent cone to a positive current

TL;DR

This paper proves the existence of the directional tangent cone to a positive current under certain conditions, extending the understanding of the geometric structure of currents in complex analysis.

Contribution

It establishes the existence of the directional tangent cone for positive currents with specific conditions on their associated currents, including the case when the current is closed.

Findings

Existence of the strict transform of a positive current under certain conditions.

Conditions for the existence of the directional tangent cone.

Additional condition for closed currents.

Abstract

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.