Non-pluripolar energy and the complex Monge-Ampère operator

Abstract

Given a domain $\Omega \subset {\mathbf{C}}^n$ we introduce a class of plurisubharmonic (psh) functions $\mathcal{G}(\Omega )$ and Monge-Ampère operators $u\mapsto [d{d}^{c}u{]}^p$, $p\le n$, on $\mathcal{G}(\Omega )$ that extend the Bedford-Taylor-Demailly Monge-Ampère operators. Here $[d{d}^{c}u{]}^p$ is a closed positive current of bidegree $(p,p)$ that dominates the non-pluripolar Monge-Ampère current $⟨d{d}^{c}u{⟩}^p$. We prove that $[d{d}^{c}u{]}^p$ is the limit of Monge-Ampère currents of certain natural regularizations of $u$. On a compact Kähler 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-Ampère currents $[d{d}^c\phi +\omega {]}^p$ for $\phi \in \mathcal{G}(X,\omega )$ that only depend on the current $d{d}^c\phi +\omega$. The limits of Monge-Ampère currents of certain natural regularizations of $\phi$ can be expressed in terms of $[d{d}^c\phi +\omega {]}^j$ for $j\le p$. We get a mass formula involving the currents $[d{d}^c\phi +\omega {]}^p$ that describes the loss of mass of the non-pluripolar Monge-Ampère measure $⟨d{d}^c\phi +\omega {⟩}^n$. 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.

38 pages