A note on the structure of plurifinely open sets and the equality of some complex Monge-Ampère measures

2023/09/06 by Mohamed El Kadiri, Kadiri, Mohamed El
Mathematics · #31C35 #31C40 #31D05 #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2309.03391

openalex publication_date 2023/09/06 · openalex created_date 2023/09/09 · openalex updated_date 2026/07/28

Abstract

In a recent preprint published on arXiv (see arXiv:2308.02993v2, referred here as \citeNXH), N.X. Hong stated that every plurifinely open set U⊂ ℂⁿ, n≥ 1, is of the form U=\bigcup \φ_j>-1\, where each ϕ_j is a negative plurisubharmonic function defined on an open ball B_j⊂ ℂⁿ and used this result to prove an equality result on complex Monge-Ampère measures. Unfortunately, this result is wrong as we will see below.

Related

• Plurifinely open sets and complex Monge-Ampère measures
• A note on equality on plurifinely open sets of some complex Monge-Ampère measures
• Plurifinely Plurisubharmonic functions and the Monge Ampère Operator
• An equality of Monge-Ampère measures
• Monge-Ampere measures on pluripolar sets