Equivalence of Steinness and Validity of Oka's principle for subdomains fo Stein manifolds
✍ Scribed by Jürgen Leiterer
- Publisher
- John Wiley and Sons
- Year
- 1979
- Tongue
- English
- Weight
- 188 KB
- Volume
- 89
- Category
- Article
- ISSN
- 0025-584X
No coin nor oath required. For personal study only.
✦ Synopsis
Equivalence of Steinness and Validity of Oka's principle for subdomains of Stein manifolds
Dedicated to the 30th Anniversary of the G.D.R. By JUR~EN LEITERER (Berlin, GDR) (Eingegangen am 20.9.1978) K. OKA [ll] proved in 1939 that every continuously solvable COUSIN-I1 problem on a domain of holomorphy is also holomorphically solvable. This principle characterizes analytic properties topologically and was called by J.-P. SERRE OKA'S principle. I n 1957 H. GRAUERT [5] obtained a profound generalization of this principle, which asserts, in particular, that the holomorphic and the topologic classifications of vector bundles over a STEIN space coincide. In future H. GRAUERT'S result was developed in several directions (see, for example, [12], [4], [2], [8], [6], [lo]). I n the present paper we investigate the converse problem whether from the validity of OKA'B principle follows Smmmess or not. As it was remarked in [7], [9], this question was recently posed by K. SHIGA to J. KAJIWARA. A similar question with respect to punctured polydiscs recently was posed by V. P. PALAMODOV to the author. I n this paper M is a STEIN manifold, D is a domain in M , and O(U) is the sheaf of germs of local holomorphic (continuous) functions on M . I f is a complex LIE group, then we denote by OG(VG) the sheaf of germs of local holomorphic (continuous) a-valued