Lelong-Demailly Numbers in Terms of Capacity and Weak Convergence for Closed Positive Currents

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• 作者：Fredj ELKHADHRA
• 关键词：positive current ; capacity
• 刊名：Acta Mathematica Scientia
• 出版年：2013
• 出版时间：November, 2013
• 年：2013
• 卷：33
• 期：6
• 页码：1652-1666
• 全文大小：315 K

文摘

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) = ¡Ò_K T ¡Ä (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 ¡ú uT, for positive closed currents T, T_k and psh functions u_k u.