Abstract
Let K-p be the set of (p, p) closed positive and conic currents on C-n such that the Lelong number is equal to 1 at the origine O, the convex subset K-p is weakly compact metrizable subspace of the space of currents on C-n. For every current T on C-n, the set of limits of the family (h(r))T-*)(r<1) is connex and weakly compact in m.K-p where m is the Lelong number of T at O. Conversely we prove in this work that if M is any closed connex subset of K-p (n-1 greater than or equal to p greater than or equal to 1); there exists a closed positive current T such that M is the limit set of the family ((h(r))T-*)(r<1). The case p=1 was first treated by C. O. Kiselman [5], and then refined by the author in collaboration with J. P. Demailly and M. Mouzali [1] to get precise quantitative estimates. In the second part we give a sufficient condition relative to the trace measure for the existence of the tangent cone of a closed positive current of type (p, p) with p=1 or n-1, We prove that this condition is not valid for the case 2 less than or equal to p less than or equal to n-2.