关键词: positive current, capacity

Abstract:

In this paper we give a new definition of the Lelong-Demailly number in terms of the C_T -capacity, where T is a closed positive current and C_T is the capacity associated to T. This derived from some esimate on the growth of the C_T -capacity of the sublevel sets of a weighted plurisubharmonic (psh for short) function. These estimates enable us to give
another proof of the Demailly´s comparaison theorem as well as a generalization of some results due to Xing concerning the characterization of bounded psh functions. Another problem that we consider here is related to the existence of a psh function v that satisfies the equality C_T (K) = ∫_K T ^ (dd^cv)^p, where K is a compact subset. Finally, we give some conditions on the capacity C_T that guarantees the weak convergence u_kT_k → uT, for positive closed currents T, T_k and psh functions u_k, u.

Key words: positive current, capacity