On the tangent cones to plurisubharmonic currents

TL;DR

This paper investigates conditions under which tangent cones exist for positive plurisubharmonic currents, providing growth estimates of Lelong functions and multiple proofs to establish their existence.

Contribution

It introduces new growth estimates for Lelong functions and offers a second proof for tangent cone existence under specific conditions.

Findings

Growth estimates of Lelong functions are established.

Existence of tangent cones is proven under certain conditions.

A second proof method for tangent cone existence is provided.

Abstract

In this paper, we study the existence of the tangent cone to a positive plurisubharmonic or plurisuperharmonic current with a suitable condition. Some Estimates of the growth of the Lelong functions associated to the current and to its $dd^c$ are given to ensure the existence of the blow-up of this current. A second proof for the existence of the tangent cone is derived from these estimates.