Entire pluricomplex Green functions and Lelong numbers of projective currents

Abstract

Let $T$ be a positive closed current of bidimension (1,1) and unit mass on the complex projective space ${\mathbb{P}}^n$. We prove that the set $V_{\alpha }(T)$ of points where $T$ has Lelong number larger than $\alpha$ is contained in a complex line if $\alpha \ge 2/3$, and $|V_{\alpha }(T)\setminus L|\le 1$ for some complex line $L$ if $1/2\le \alpha <2/3$. We also prove that in dimension 2 and if $2/5\le \alpha <1/2$, then $|V_{\alpha }(T)\setminus C|\le 1$ for some conic $C$.

9 pages