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Theorems on the classical problems in the theory of functions of several complex variables, first proved by K. Oka between 1930 and 1950 (see [[#References|[1]]]).
 
Theorems on the classical problems in the theory of functions of several complex variables, first proved by K. Oka between 1930 and 1950 (see [[#References|[1]]]).
  
1) Oka's theorem on the [[Cousin problems|Cousin problems]]: The first Cousin problem is solvable in any [[Domain of holomorphy|domain of holomorphy]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068140/o0681401.png" />; the second Cousin problem is solvable in any domain of holomorphy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068140/o0681402.png" /> that is homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068140/o0681403.png" />, where all domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068140/o0681404.png" />, except for, possibly, one, are simply connected.
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1) Oka's theorem on the [[Cousin problems|Cousin problems]]: The first Cousin problem is solvable in any [[Domain of holomorphy|domain of holomorphy]] in $\mathbf C^n$; the second Cousin problem is solvable in any domain of holomorphy $D\subset\mathbf C^n$ that is homeomorphic to $D_1\times\dotsb\times D_n$, where all domains $D_v\subset\mathbf C$, except for, possibly, one, are simply connected.
  
 
2) Oka's theorem on the [[Levi problem|Levi problem]]: Any pseudo-convex Riemannian domain (cf. [[Pseudo-convex and pseudo-concave|Pseudo-convex and pseudo-concave]]) is a domain of holomorphy.
 
2) Oka's theorem on the [[Levi problem|Levi problem]]: Any pseudo-convex Riemannian domain (cf. [[Pseudo-convex and pseudo-concave|Pseudo-convex and pseudo-concave]]) is a domain of holomorphy.
  
Originally Oka proved these theorems in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068140/o0681405.png" />; in the case of arbitrary dimension, the theorems were also proved by other mathematicians.
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Originally Oka proved these theorems in dimension $n=2$; in the case of arbitrary dimension, the theorems were also proved by other mathematicians.
  
3) The Oka–Weil theorem: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068140/o0681406.png" /> be a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068140/o0681407.png" /> and let the compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068140/o0681408.png" /> coincide with its hull with respect to the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068140/o0681409.png" /> of all functions holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068140/o06814010.png" /> (cf. [[Holomorphic envelope|Holomorphic envelope]]); then for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068140/o06814011.png" /> holomorphic in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068140/o06814012.png" />, and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068140/o06814013.png" />, a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068140/o06814014.png" /> can be found such that
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3) The Oka–Weil theorem: Let $D$ be a domain in $\mathbf C^n$ and let the compact set $K\subset D$ coincide with its hull with respect to the algebra $\mathcal O(D)$ of all functions holomorphic in $D$ (cf. [[Holomorphic envelope|Holomorphic envelope]]); then for any function $f$ holomorphic in a neighbourhood of $K$, and for any $\epsilon>0$, a function $F\in\mathcal O(D)$ can be found such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068140/o06814015.png" /></td> </tr></table>
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$$\max_K|f-F|<\epsilon.$$
  
 
This fundamental theorem in the theory of holomorphic approximation is extensively used in complex and functional analysis.
 
This fundamental theorem in the theory of holomorphic approximation is extensively used in complex and functional analysis.
  
4) Oka's coherence theorem: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068140/o06814016.png" /> be a sheaf of holomorphic functions on a complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068140/o06814017.png" />; then for any natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068140/o06814018.png" />, any locally finitely-generated subsheaf of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068140/o06814019.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068140/o06814020.png" /> times) is a coherent [[Analytic sheaf|analytic sheaf]] (cf. [[Coherent sheaf|Coherent sheaf]]).
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4) Oka's coherence theorem: Let $\mathcal O$ be a sheaf of holomorphic functions on a complex manifold $X$; then for any natural number $p$, any locally finitely-generated subsheaf of the sheaf $\mathcal O^p=\mathcal O\times\dotsb\times\mathcal O$ ($p$ times) is a coherent [[Analytic sheaf|analytic sheaf]] (cf. [[Coherent sheaf|Coherent sheaf]]).
  
This is one of the basic theorems of the so-called Oka–Cartan theory, which is used essentially in proving the Cartan theorems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068140/o06814021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068140/o06814022.png" /> (cf. [[Cartan theorem|Cartan theorem]]).
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This is one of the basic theorems of the so-called Oka–Cartan theory, which is used essentially in proving the Cartan theorems $A$ and $B$ (cf. [[Cartan theorem|Cartan theorem]]).
  
 
====References====
 
====References====

Latest revision as of 13:49, 14 February 2020

Theorems on the classical problems in the theory of functions of several complex variables, first proved by K. Oka between 1930 and 1950 (see [1]).

1) Oka's theorem on the Cousin problems: The first Cousin problem is solvable in any domain of holomorphy in $\mathbf C^n$; the second Cousin problem is solvable in any domain of holomorphy $D\subset\mathbf C^n$ that is homeomorphic to $D_1\times\dotsb\times D_n$, where all domains $D_v\subset\mathbf C$, except for, possibly, one, are simply connected.

2) Oka's theorem on the Levi problem: Any pseudo-convex Riemannian domain (cf. Pseudo-convex and pseudo-concave) is a domain of holomorphy.

Originally Oka proved these theorems in dimension $n=2$; in the case of arbitrary dimension, the theorems were also proved by other mathematicians.

3) The Oka–Weil theorem: Let $D$ be a domain in $\mathbf C^n$ and let the compact set $K\subset D$ coincide with its hull with respect to the algebra $\mathcal O(D)$ of all functions holomorphic in $D$ (cf. Holomorphic envelope); then for any function $f$ holomorphic in a neighbourhood of $K$, and for any $\epsilon>0$, a function $F\in\mathcal O(D)$ can be found such that

$$\max_K|f-F|<\epsilon.$$

This fundamental theorem in the theory of holomorphic approximation is extensively used in complex and functional analysis.

4) Oka's coherence theorem: Let $\mathcal O$ be a sheaf of holomorphic functions on a complex manifold $X$; then for any natural number $p$, any locally finitely-generated subsheaf of the sheaf $\mathcal O^p=\mathcal O\times\dotsb\times\mathcal O$ ($p$ times) is a coherent analytic sheaf (cf. Coherent sheaf).

This is one of the basic theorems of the so-called Oka–Cartan theory, which is used essentially in proving the Cartan theorems $A$ and $B$ (cf. Cartan theorem).

References

[1] K. Oka, "Sur les fonctions analytiques de plusieurs variables" , Iwanami Shoten (1961)
[2] L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973)
[3] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965)


Comments

Reference [a2] is an annotated English translation of Oka's fundamental papers.

References

[a1] S.G. Krantz, "Function theory of several complex variables" , Wiley (1982)
[a2] K. Oka, "Collected papers" , Springer (1984)
[a3] R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) pp. Chapt. 1, Sect. 3
How to Cite This Entry:
Oka theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Oka_theorems&oldid=19092
This article was adapted from an original article by E.M. Chirka (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article