Non-pluripolar energy and the complex Monge-Amp\`ere operator

Abstract

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