Entire pluricomplex Green functions and Lelong numbers of projective currents

Abstract

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