Paper detail

Monge-Ampère measures on contact sets

Let $(X, ω)$ be a compact Kähler manifold of complex dimension n and $θ$ be a smooth closed real $(1,1)$-form on $X$ such that its cohomology class $\{ θ\}\in H^{1,1}(X, \mathbb{R})$ is pseudoeffective. Let $φ$ be a $θ$-psh function, and let $f$ be a continuous function on $X$ with bounded distributional laplacian with respect to $ω$ such that $φ\leq f. $ Then the non-pluripolar measure $θ_φ^n:= (θ+ dd^c φ)^n$ satisfies the equality: $$ {\bf{1}}_{\{ φ= f \}} \ θ_φ^n = {\bf{1}}_{\{ φ= f \}} \ θ_f^n,$$ where, for a subset $T\subseteq X$, ${\bf{1}}_T$ is the characteristic function. In particular we prove that \[ θ_{P_θ(f)}^n= { \bf {1}}_{\{P_θ(f) = f\}} \ θ_f^n\qquad {\rm and }\qquad θ_{P_θ[φ](f)}^n = { \bf {1}}_{\{P_θ[φ](f) = f \}} \ θ_f^n. \]