# 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 |

#### Comments

#### References

[a1] | S. Kobayashi, H. Wu, "Complex differential geometry" , Birkhäuser (1983) |

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Levi problem.

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