Возрастающее объединение пространств Стейна с сингулярностями

We show that if $X$ is a Stein space and, if $\Omega\subset X$ is exhaustable by a sequence $\Omega_{1}\subset\Omega_{2}\subset\ldots\subset\Omega_{n}\subset\dots$ of open Stein subsets of $X$, then $\Omega$ is Stein. This generalizes a well-known result of Behnke and Stein which is obtained for $X=\mathbb{C}^{n}$ and solves the union problem, one of the most classical questions in Complex Analytic Geometry. When $X$ has dimension $2$, we prove that the same result follows if we assume only that $\Omega\subset\subset X$ is a domain of holomorphy in a Stein normal space. It is known, however, that if $X$ is an arbitrary complex space which is exhaustable by an increasing sequence of open Stein subsets $X_{1}\subset X_{2}\subset\dots\subset X_{n}\subset\dots$, it does not follow in general that $X$ is holomorphically-convex or~holomorphically-separate (even if $X$ has no singularities). One can even obtain $2$-dimensional complex manifolds on which all holomorphic functions are constant.

Bergman–Einstein metrics, a generalization of Kerner’s theorem and Stein spaces with spherical boundaries

We give an affirmative solution to a conjecture of Cheng proposed in 1979 which asserts that the Bergman metric of a smoothly bounded strongly pseudoconvex domain in {\mathbb{C}^{n},n\geq 2}, is Kähler–Einstein if and only if the domain is biholomorphic to the ball. We establish a version of the classical Kerner theorem for Stein spaces with isolated singularities which has an immediate application to construct a hyperbolic metric over a Stein space with a spherical boundary.

Some properties of Grauert type surfaces

In a previous work, we classified weakly complete surfaces which admit a real analytic plurisubharmonic exhaustion function; we showed that, if they are not proper over a Stein space, then they admit a pluriharmonic function, with compact Levi-flat level sets foliated with dense complex leaves. We called these Grauert type surfaces. In this note, we investigate some properties of these surfaces. Namely, we prove that the only compact curves that can be contained in them are negative in the sense of Grauert and that the level sets of the pluriharmonic function are connected, thus completing the analogy with the Cartan–Remmert reduction of a holomorphically convex space. Moreover, in our classification theorem, we had to pass to a double cover to produce the pluriharmonic function; the last part of the present paper is devoted to the construction of an example where passing to a double cover cannot be avoided.

Dense holomorphic curves in spaces of holomorphic maps and applications to universal maps

We study when there exists a dense holomorphic curve in a space of holomorphic maps from a Stein space. We first show that for any bounded convex domain [Formula: see text] and any connected complex manifold [Formula: see text], the space [Formula: see text] contains a dense holomorphic disc. Our second result states that [Formula: see text] is an Oka manifold if and only if for any Stein space [Formula: see text] there exists a dense entire curve in every path component of [Formula: see text]. In the second half of this paper, we apply the above results to the theory of universal functions. It is proved that for any bounded convex domain [Formula: see text], any fixed-point-free automorphism of [Formula: see text] and any connected complex manifold [Formula: see text], there exists a universal map [Formula: see text]. We also characterize Oka manifolds by the existence of universal maps.

On thickening of holomorphic hulls and envelopes of holomorphy on Stein spaces

On a normal Stein variety [Formula: see text], we study the thickening problem, i.e. the problem whether the assumption that a compact set [Formula: see text] is contained in the interior of another compact set, [Formula: see text] implies that the same inclusion holds for their holomorphic hulls. An affirmative answer is given for [Formula: see text] with isolated quotient singularities. On any Stein space [Formula: see text] with isolated singularities, we prove thickening for those hulls which have analytic structure at the singular points, obtaining a limitation for possible counter-examples. In dimension [Formula: see text], we finally relate the holomorphic hulls to analytic extension from parts of strictly pseudoconvex boundaries.

Strong disk property for domains in open Riemann surfaces

We study the relation between the holomorphic approximation property and the strong disk property for an open set of an open Riemann surface or a Stein space of pure dimension 1.

On Hartogs extension theorems for separately (⋅, W)-holomorphic functions

In this paper, we study the holomorphic extension of separately (⋅, W)-holomorphic functions from a product of a [Formula: see text]-regular compact subset in a Stein space with a Stein space to some its neighborhood. At the same time, we generalize the Siciak's result to separately (⋅, W)-holomorphic functions with pluripolar singularities on the crosses.