Monge-Ampère measures on contact sets

Abstract

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 $\{\theta \}\in H^{1,1}(X,\mathbb{R})$ is pseudoeffective. Let $\phi$ be a $\theta$-psh function, and let $f$ be a continuous function on $X$ with bounded distributional laplacian with respect to $\omega$ such that $\phi \le f.$ Then the non-pluripolar measure ${\theta }_{\phi }^n:=(\theta +d{d}^c\phi {)}^n$ satisfies the equality: $$1_{\{\phi =f\}}{\theta }_{\phi }^n=1_{\{\phi =f\}}{\theta }_f^n,$$ where, for a subset $T\subseteq X$, $1_T$ is the characteristic function. In particular we prove that \[ θ_{P_θ(f)}^n= { \bf {1}}_{\{P_θ(f) = f\}} \ θ_f^n\qquad {\rm and }\qquad θ_{P_θ[φ](f)}^n = { \bf {1}}_{\{P_θ[φ](f) = f \}} \ θ_f^n. \]

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