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Monge-Amp\`{e}re measures on contact sets

Let $(X, \omega)$ be a compact K\"ahler 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 $\varphi$ be a $\theta$-psh function, and let $f$ be a...

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Bibliographic Details
Published in:	arXiv.org 2019-12
Main Authors:	Eleonora Di Nezza, Trapani, Stefano
Format:	Article
Language:	English
Subjects:	Characteristic functions Continuity (mathematics) Homology
Online Access:	Get full text
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Summary:	Let $(X, \omega)$ be a compact K\"ahler 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 $\varphi$ be a $\theta$-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 $\theta_\varphi^n:= (\theta + dd^c \varphi)^n$ satisfies the equality: $$ {\bf{1}}_{\{ \varphi = f \}} \ \theta_\varphi^n = {\bf{1}}_{\{ \varphi = f \}} \ \theta_f^n,$$ where, for a subset $T\subseteq X$, ${\bf{1}}_T$ is the characteristic function. In particular we prove that \[ \theta_{P_{\theta}(f)}^n= { \bf {1}}_{\{P_{\theta}(f) = f\}} \ \theta_f^n\qquad {\rm and }\qquad \theta_{P_\theta[\varphi](f)}^n = { \bf {1}}_{\{P_\theta[\varphi](f) = f \}} \ \theta_f^n. \]
ISSN:	2331-8422

Summary:	Let \((X, \omega)\) be a compact K\"ahler 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 \(\varphi\) be a \(\theta\)-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 \(\theta_\varphi^n:= (\theta + dd^c \varphi)^n\) satisfies the equality: $$ {\bf{1}}_{\{ \varphi = f \}} \ \theta_\varphi^n = {\bf{1}}_{\{ \varphi = f \}} \ \theta_f^n,$$ where, for a subset \(T\subseteq X\), \({\bf{1}}_T\) is the characteristic function. In particular we prove that \[ \theta_{P_{\theta}(f)}^n= { \bf {1}}_{\{P_{\theta}(f) = f\}} \ \theta_f^n\qquad {\rm and }\qquad \theta_{P_\theta[\varphi](f)}^n = { \bf {1}}_{\{P_\theta[\varphi](f) = f \}} \ \theta_f^n. \]
ISSN:	2331-8422