# 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