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 $\alpha \geq 2/3$, and $|V_\alpha (T)\setminus L|\leq 1$ for some complex line $L$ if $\alpha \geq 1/2$. We also prove that in dimension 2 and if $\alpha \geq 2/5$, then $|V_\alpha (T)\setminus C|\leq 1$ for some conic $C$.