Abstract
We give two examples of complex spaces on which global holomorphic functions separate points and give local coordinates and they cannot be realized as open subsets of Stein spaces. At the same time we notice that these examples are open subsets of Stein schemes, a notion introduced by Grauert (Math Z 81:377–391, 1963). In the context of complex schemes we notice that by contracting a Nori string one obtains a complex scheme and not a complex space. The covering spaces of 1-convex surfaces are divided in two categories: those that have an envelope of holomorphy and those that do not. More interesting are those in the second category and they correspond to covering spaces for singularities which in the desingularization with normal crossings contain cycles in the exceptional set.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brudnyi, A.: On holomorphic \(L^2\) functions on coverings of strongly pseudoconvex manifolds. Publ. Res. Inst. Math. Sci. 43(4), 963–976 (2007)
Colţoiu, M.: Some open problems concerning Stein spaces. Analyse complexe (Bucharest, 1989). Rev. Roumaine Math. Pures Appl. 36(5–6), 225–229 (1991)
Colţoiu, M., Diederich, K.: Open sets with Stein hypersurface sections in Stein spaces. Ann. Math. 145(1–2), 175–182 (1997)
Colţoiu, M., Diederich, K.: On Levi’s problem on complex spaces and envelopes of holomorphy. Math. Ann. 316(1), 185–199 (2000)
Colţoiu, M., Tibăr, M.: Steinness of the universal covering of the complement of a 2-dimensional complex singularity. Math. Ann. 326(1), 95–104 (2003)
Falbel, E.: Nonembeddable CR-manifolds and surface singularities. Invent. Math. 108(1), 49–65 (1992)
Grauert, H.: Bemerkenswerte pseudokonvexe Mannigfaltigkeiten. Math. Z. 81, 377–391 (1963)
Joiţa, C.: On uniformly Runge domains. J. Math. Kyoto Univ. 47(4), 875–880 (2007)
Lieb, I.: Über komplexe Räume and komplexe Spektren. Invent. Math. 1, 45–58 (1966)
Narasimhan, R.: Several complex variables. Chicago Lectures in Mathematics. The University of Chicago Press, Chicago (1971)
Suzuki, O.: A new class of domains of holomorphy. I. The concepts of boundary resolutions and \(L\)-manifolds. Publ. Res. Inst. Math. Sci. 13(2), 497–521 (1977/1978)
Acknowledgments
Both authors were partially supported by CNCS grant PN-II-ID-PCE-2011-3-0269.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of Professor Hans Grauert.