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 ${\Bbb P}^n$. 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 $\alpha\geq2/3$, and $|V_\alpha(T)\setminus L|\leq1$ for some complex line $L$ if $1/2\leq\alpha<2/3$. We also prove that in dimension 2 and if $2/5\leq\alpha<1/2$, then $|V_\alpha(T)\setminus C|\leq1$ for some conic $C$.