LELONG-DEMAILLY NUMBERS IN TERMS OF CAPACITY AND WEAK CONVERGENCE FOR CLOSED POSITIVE CURRENTS

摘要

In this paper we give a new definition of the Lelong-Demailly number in terms of the CT-capacity,where T is a closed positive current and CT is the capacity associated to T.This derived from some esimate on the growth of the CT-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 CT(K)=∫KT∧(ddcv)p,where K is a compact subset.Finally,we give some conditions on the capacity CT that guarantees the weak convergence ukTk → uT,for positive closed currents T,Tk and psh functions uk,u.