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The problem of the geometric characterization of domains in a given [[Analytic space|analytic space]] that are Stein spaces (cf. [[Stein space|Stein space]]); it was posed by E.E. Levi [[#References|[1]]] for domains in the affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l0582601.png" /> in the following form. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l0582602.png" /> be a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l0582603.png" /> each boundary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l0582604.png" /> of which has the following property: there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l0582605.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l0582606.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l0582607.png" /> and a holomorphic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l0582608.png" /> that cannot be extended holomorphically to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l0582609.png" />. Is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826010.png" /> a [[Domain of holomorphy|domain of holomorphy]]? This property is equivalent to any of the following assertions about the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826011.png" />: 1) for no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826012.png" /> is there a sequence of bounded holomorphic surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826013.png" /> that converges to a holomorphic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826014.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826017.png" />; 2) the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826018.png" /> is pseudo-convex, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826021.png" /> is the Euclidean distance, is a [[Plurisubharmonic function|plurisubharmonic function]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826022.png" />; and 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826023.png" /> is a pseudo-convex manifold, that is, there is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826024.png" /> a plurisubharmonic function that tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826025.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826026.png" /> is approached. The Levi problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826027.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826029.png" /> (cf. [[Covering domain|Covering domain]]) (see –[[#References|[6]]]). Oka's result has been generalized to domains spread over any Stein manifold: If such a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826030.png" /> is a pseudo-convex manifold, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826031.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826032.png" /> 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 [[#References|[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|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 , [[#References|[4]]] and also [[Modification|Modification]]; [[Proper morphism|Proper morphism]]).
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The Levi problem can also be posed for domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826033.png" /> in an infinite-dimensional complex topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826034.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826035.png" /> is locally convex and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826036.png" /> is a domain of holomorphy, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826037.png" /> is pseudo-convex, that is, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826038.png" /> there is a plurisubharmonic function that tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826039.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826040.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058260/l05826041.png" /> (see ).
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The problem of the geometric characterization of domains in a given [[Analytic space|analytic space]] that are Stein spaces (cf. [[Stein space|Stein space]]); it was posed by E.E. Levi [[#References|[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|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|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|Covering domain]]) (see –[[#References|[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 [[#References|[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|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 , [[#References|[4]]] and also [[Modification|Modification]]; [[Proper morphism|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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.C. Gunning,  H. Rossi,  "Analytic functions of several complex variables" , Prentice-Hall  (1965)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  B.A. Fuks,  "Special chapters in the theory of analytic functions of several complex variables" , Amer. Math. Soc.  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''2''' , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  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</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.C. Gunning,  H. Rossi,  "Analytic functions of several complex variables" , Prentice-Hall  (1965)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  B.A. Fuks,  "Special chapters in the theory of analytic functions of several complex variables" , Amer. Math. Soc.  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''2''' , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  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</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,  H. Wu,  "Complex differential geometry" , Birkhäuser  (1983)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,  H. Wu,  "Complex differential geometry" , Birkhäuser  (1983)</TD></TR></table>

Revision as of 22:16, 5 June 2020


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)
How to Cite This Entry:
Levi problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Levi_problem&oldid=47620
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article