2004/11/04 by Sarkis Frederic, Sarkis Frédéric, Frederic, Sarkis
Mathematics · #32Q55 #32d10 #Complex Variables (math.CV) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Holomorphic and Operator Theory #math.CV #msc:32Q55 #msc:32d10
paper · pdf · doi:10.48550/arxiv.math/0411083
9 pages, 1 figure
arxiv created 2004/11/04 · openalex publication_date 2004/11/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A question of Poletsky was to know if there exists a thin Hartogs figure such that any of its neighborhoods cannot be imbedded in Stein spaces. In \citechirka, Chirka and Ivashkovitch gave such an example arising in an open complex manifold. In this paper, we answer to the question of the existence of such a figure in compact surfaces by giving an example arising in P2(ℂ). By smoothing it, we obtain a smooth (non analytic) disc with boundary D ⊂ P2(ℂ) having the same property. Consequently, this disc intersects all algebraic curves of P2(ℂ). Moreover, as D is topologically trivial, it has a neighborhood diffeomorphic to the unit ball of ℂ2. This gives a negative answer to the following question of S. Ivashkovitch: Is the property for a domain B of P2(ℂ) to be diffeormorphic to the unit ball of ℂ2 a sufficient condition for the existence of non-constant holomorphic functions on it?