On nonimbeddability of topologically trivial domains and Thin Hartogs figures of $P_2(\mathbb{C})$ into Stein spaces

Abstract

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 \cite{chirka}, 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 $P_2(\mathbb{C})$. By smoothing it, we obtain a smooth (non analytic) disc with boundary $\bar{D} \subset P_2(\mathbb{C})$ having the same property. Consequently, this disc intersects all algebraic curves of $P_2(\mathbb{C})$. Moreover, as $\bar D$ is topologically trivial, it has a neighborhood diffeomorphic to the unit ball of $\mathbb{C}^2$. This gives a negative answer to the following question of S. Ivashkovitch: Is the property for a domain $B$ of $P_2(\mathbb{C})$ to be diffeormorphic to the unit ball of $\mathbb{C}^2$ a sufficient condition for the existence of non-constant holomorphic functions on it?