Characterizing the range of the complex Monge-Ampère operator

TL;DR

This work extends the solvability of the complex Monge–Ampère equation to measures with pluripolar parts on compact Kähler manifolds by globalizing the Cegrell–Lebesgue decomposition. It leverages the non-pluripolar energy framework, the Blocki–Cegrell class $\mathcal{D}(X,\omega)$, and the relative full mass theory to decompose $\mu=\mu_r+\mu_s$ with $\mu_r=f\omega^n$, $f\in L^p(\omega^n)$, $p>1$, and $\mu_s$ pluripolar, constructing solutions $\psi\in\mathcal{D}(X,\omega)$ to $MA_\omega(\psi)=\mu$ whenever the singular part is realized by a model potential. The paper also provides a mixed Monge–Ampère formula on compact Kähler surfaces and demonstrates a concrete path to solutions for convex combinations $ (1-t) f\omega^n + t\mu$. These results broaden global pluripotential theory by enabling PDEs with pluripolar data in the $DMA\cap \mathcal{G}$ setting, with practical implications for constructing singular Kähler metrics and understanding degenerations of complex structures.

Abstract

In this note, we solve the complex Monge-Ampère equation for measures with a pluripolar part in compact Kähler manifolds. This result generalizes the classical results obtained by Cegrell in bounded hyperconvex domains. We also discuss the properties of the complex Monge-Ampère operator in some special cases.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• Definition 2.2
• Proposition 2.3
• Definition 2.4
• Definition 2.5
• Lemma 2.6