Levi problem

The problem of the geometric characterization of domains in a given analytic space that are Stein spaces (cf. Stein space); it was posed by E.E. Levi [1] for domains in the affine space $ \mathbf C ^ {n} $ in the following form. Let $ D $ be a domain in $ \mathbf C ^ {n} $ each boundary point $ \zeta $ of which has the following property: there is a neighbourhood $ U $ of $ \zeta $ in $ \mathbf C ^ {n} $ and a holomorphic function in $ U \cap D $ that cannot be extended holomorphically to $ \zeta $. Is $ D $ a domain of holomorphy? This property is equivalent to any of the following assertions about the domain $ D $: 1) for no $ \zeta \in \partial D $ is there a sequence of bounded holomorphic surfaces $ S _ \nu $ that converges to a holomorphic surface $ S $ with $ \partial S _ \nu \rightarrow \partial S $, $ \overline{S}\; _ \nu , \partial S \subset D $, $ \zeta \in S $; 2) the domain $ D $ is pseudo-convex, that is, $ - \mathop{\rm log} \rho ( z , \partial D ) $, $ z \in D $, where $ \rho $ is the Euclidean distance, is a plurisubharmonic function in $ D $; and 3) $ D $ is a pseudo-convex manifold, that is, there is in $ D $ a plurisubharmonic function that tends to $ + \infty $ as $ \partial D $ is approached. The Levi problem for $ \mathbf C ^ {n} $ was affirmatively solved in 1953–1954 independently by K. Oka, H. Bremermann and F. Norguet, and Oka solved the problem in a more general formulation, concerned with domains spread over $ \mathbf C ^ {n} $( cf. Covering domain) (see –[6]). Oka's result has been generalized to domains spread over any Stein manifold: If such a domain $ D $ is a pseudo-convex manifold, then $ D $ is a Stein manifold. The Levi problem has also been affirmatively solved in a number of other cases, for example, for non-compact domains spread over the projective space $ \mathbf C P ^ {n} $ or over a Kähler manifold on which there exists a strictly plurisubharmonic function (see ), and for domains in a Kähler manifold with positive holomorphic bisectional curvature [7]. At the same time, examples of pseudo-convex manifolds and domains are known that are not Stein manifolds and not even holomorphically convex. A necessary and sufficient condition for a complex space to be a Stein space is that it is strongly pseudo-convex (see Pseudo-convex and pseudo-concave). Also, a strongly pseudo-convex domain in any complex space is holomorphically convex and is a proper modification of a Stein space (see , [4] and also Modification; Proper morphism).

The Levi problem can also be posed for domains $ D $ in an infinite-dimensional complex topological vector space $ E $. If $ E $ is locally convex and $ D $ is a domain of holomorphy, then $ D $ is pseudo-convex, that is, in $ D $ there is a plurisubharmonic function that tends to $ + \infty $ as $ \partial D $ is approached. The converse theorem is false even in Banach spaces, but it has been proved for Banach spaces with a countable basis, as well as for a number of other classes of spaces $ E $( see ).

References

[1]	E.E. Levi, "Sulle superficie dello spazio a 4 dimensione che possono essere frontiera del campo di esistenza di una funzione analitica di due variabili complesse" Ann. Mat. Pura Appl. , 18 (1911) pp. 69–79
[2a]	A.L. Onishchik, "Stein spaces" J. Soviet Math. , 4 : 5 (1975) pp. 540–554 Itogi Nauk. i Tekhn. Algebra Topol. Geom. , 11 (1974) pp. 125–151
[2b]	A.L. Onishchik, "Pseudoconvexity in the theory of complex spaces" J. Soviet Math. , 14 : 4 (1980) pp. 1363–1407 Itogi Nauk. i Tekhn. Algebra Topol. Geom. , 15 (1977) pp. 93–171
[3]	V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)
[4]	R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965)
[5]	B.A. Fuks, "Special chapters in the theory of analytic functions of several complex variables" , Amer. Math. Soc. (1965) (Translated from Russian)
[6]	B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian)
[7]	O. Suzuki, "Pseudoconvex domains on a Kähler manifold with positive holomorphic, bisectional curvature" Publ. Res. Inst. Math. Sci. Kyoto Univ. , 12 (1976) pp. 191–214; 439–445
[a1]	S. Kobayashi, H. Wu, "Complex differential geometry" , Birkhäuser (1983)