×

Capacity associated to a positive closed current. (Capacité associée à un courant positif fermé.) (French. English summary) Zbl 1114.31004

Summary: Let \(\Omega\) be an open set of \(\mathbb{C}^n\) and \(T\) be a positive closed current of dimension \(p\geq 1\) on \(\Omega\), we define a capacity associated to \(T\) by: \[ C_T(K,\Omega)=C_T(K)=\sup \left\{ds\int_K{T\wedge(dd^c v)^p,\;v\in \text{psh}(\Omega),\;0<v<1}\right\}, \] where \(K\) is a compact set of \(\Omega\). We prove, in the same way as Bedford-Taylor, that a locally bounded plurisubharmonic function is quasi-continuous with respect to \(C_T\). In the second part we define the convergence relatively to \(C_T\) and we prove that if \((u_j)\) is a family of locally uniformly bounded plurisubharmonic functions and \(u\) is a locally bounded plurisubharmonic function such that \(u_j \rightarrow u\) relatively to \(C_T\) then \(T\wedge (dd^cu_j)^p\rightarrow T\wedge (dd^cu)^p\) in the current sense.

MSC:

31C15 Potentials and capacities on other spaces
31C10 Pluriharmonic and plurisubharmonic functions
32C30 Integration on analytic sets and spaces, currents
32W20 Complex Monge-Ampère operators
PDFBibTeX XMLCite
Full Text: EuDML EMIS