Monge–Ampère measures on contact sets

Let $(X, \omega)$ be a compact Kähler manifold of complex dimension $n$ and $\theta$ be a smooth closed real $(1,1)$-form on $X$ such that its cohomology class ${\lbrace \theta \rbrace} \in H^{1,1} (X,\mathbf{R})$ is pseudo-effective. Let $\varphi$ be a $\theta \textrm{-psh}$ function, and let $f$ be a continuous function on $X$ with bounded distributional laplacian with respect to $\omega$ such that $\varphi \leq f$. Then the non-pluripolar measure $\theta^n_\varphi := (\theta + dd^c \varphi)^n$ satisfies the equality:\[1_{\lbrace \varphi=f \rbrace} \theta^n_\varphi = 1_{\lbrace \varphi=f \rbrace} \theta^n_f \; \textrm{,}\]where, for a subset $T \subseteq X , 1_T$ is the characteristic function. In particular we prove that\[\theta^n_{P_\theta (f)} = 1_{\lbrace P_\theta (f)=f \rbrace} \theta^n_f\quad \textrm{and} \quad\theta^n_{\lbrace P_\theta [\varphi] (f)=f \rbrace} = 1_{\lbrace P_\theta [\varphi] (f)=f \rbrace} \: \theta^n_f \textrm{.}\]

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Received 6 January 2020

Accepted 29 March 2020

Published 16 August 2022