arXiv2022
Given a domain $Ω\subset \mathbf C^n$ we introduce a class of plurisubharmonic (psh) functions $\mathcal G(Ω)$ and Monge-Ampère operators $u\mapsto [dd^c u]^p$, $p\leq n$, on $\mathcal G(Ω)$ that extend the Bedford-Taylor-Demailly Monge-Ampère operators. Here $[dd^c u]^p$ is a closed positive current of bidegree $(p,p)$ that dominates the non-pluripolar Monge-Ampère current $\langle dd^c u\rangle^p$. We prove that $[dd^c u]^p$ is the limit of Monge-Ampère currents of certain natural regularizations of $u$. On a compact Kähler manifold $(X, ω)$ we introduce a notion of non-pluripolar energy and a corresponding finite energy class $\mathcal G(X, ω)\subset \text{PSH}(X, ω)$ that is a global version of $\mathcal G(Ω)$. From the local construction we get global Monge-Ampère currents $[dd^c φ+ ω]^p$ for $φ\in \mathcal G(X,ω)$ that only depend on the current $dd^c φ+ ω$. The limits of Monge-Ampère currents of certain natural regularizations of $φ$ can be expressed in terms of $[dd^c φ+ ω]^j$ for $j\leq p$. We get a mass formula involving the currents $[dd^c φ+ω]^p$ that describes the loss of mass of the non-pluripolar Monge-Ampère measure $\langle dd^c φ+ω\rangle^n$. The class $\mathcal G(X, ω)$ includes $ω$-psh functions with analytic singularities and the class $\mathcal E(X, ω)$ of $ω$-psh functions of finite energy and certain other convex energy classes, although it is not convex itself.