The author introduces holomorphic currents for the purpose of constructing plurisubharmonic functions on domains in \(\mathbb{C}^ n\). Let \(U\) denote the unit disk in \(\mathbb{C}\) with boundary \(S\). For a domain \(D\) in \(\mathbb{C}^ n\) let \(A(U,D)\) denote the set of all holomorphic mappings of \(U\) into \(D\) and let \(\widetilde {A}(U, D)\) denote the set of all pairs \(p = (f, Z)\), where \(f \in A(U, D)\) and \(Z\) is a subset of \(U\).
A holomorphic current is a mapping \(\Phi\) defined on a subset \(M\) of \(\widetilde {A}(U,D)\) into the set of measures on \(\overline{U}\) which are positive on \(U\) and bounded above on \(S\). Both the mapping \(\Phi\) and domain \(M\) are required to satisfy additional properties, including the requirement that the measure \(\Phi(p)\) must be invariant with respect to proper self mappings of \(U\). Each holomorphic current \(\Phi\) generates the functional \(H(p) = h_ p(0)\), where for each \(p \in M\), \(u_ p\) is the subharmonic function on \(U\) defined by \[ u_ p(z) = G_{\Phi(p) |_ U}(z) + P_{\Phi(p)|_ S} (z), \] where for measures \(\mu\) on \(U\) and \(\nu\) on \(S\), \(G_ \mu\) and \(P_ \nu\) denote the Green potential of \(\mu\) and Poisson integral of \(\nu\). One of the main results of the paper is the following:
Theorem 5.1. If \(\Phi\) is an approximately upper semicontinuous holomorphic current bounded above on compacta of a domain \(D\subset \mathbb{C}^ n\), then \(u(z) = \inf u_ p(0)\), where the infimum is taken over all \(p = (f,Z) \in M\) with \(f(0) = z\), is plurisubharmonic on \(D\).
In section 6 it is shown that for every \(p = (f, Z) \in M\), the measure \(\Phi(p) |_ U \leq \Delta(u \circ f)\), but if \(p\) is extremal at \(f(0)\), i.e., \(H(p) = u(f(0))\), then \(\Phi|_ U = \Delta(u \circ f)\). In subsequent sections the method is applied to the construction of polynomial hulls, boundary problems of plurisubharmonic functions, the theory of capacities of boundary sets, and to solving the degenerate Monge-Ampère equation.