Non-pluripolar energy and the complex Monge-Ampere operator

Journal article, 2022

Given a domain Omega subset of C-n we introduce a class of plurisubharmonic (psh) functions G(Omega) and Monge-Ampere operators u -> [dd(c)u](p), p <= n, on G(Omega) that extend the Bedford-Taylor-Demailly Monge-Ampere operators. Here [dd(c)u](p) is a closed positive current of bidegree (p, p) that dominates the non-pluripolar Monge-Ampere current < dd(c)u >(p). We prove that [dd(c)u](p) is the limit of Monge-Ampere currents of certain natural regularizations of u. On a compact Kahler manifold (X, omega) we introduce a notion of non-pluripolar energy and a corresponding finite energy class G(X, omega) subset of PSH(X, omega) that is a global version of the class G(Omega). From the local construction we get global Monge-Ampere currents [dd(c)phi + omega](p) for phi is an element of G(X, omega) that only depend on the current dd(c)phi + omega. The limits of Monge-Ampere currents of certain natural regularizations of phi can be expressed in terms of [dd(c)phi + omega](j) for j <= p. We get a mass formula involving the currents [dd(c)phi + omega](p) that describes the loss of mass of the non-pluripolar Monge-Ampere measure < dd(c)phi + omega >(n). The class G(X, omega) includes omega-psh functions with analytic singularities and the class E(X, omega) of omega-psh functions of finite energy and certain other convex energy classes, although it is not convex itself.