Non-pluripolar energy and the complex Monge–Ampère operator

Abstract

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