View source for Oka theorems

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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]]]).

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 $D_1\times\ldots\times D_n$, 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.

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|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 $\mathcal O^p=\mathcal O\times\ldots\times\mathcal O$ ($p$ 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 $A$ and $B$ (cf. [[Cartan theorem|Cartan theorem]]).

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



====Comments====
Reference [[#References|[a2]]] is an annotated English translation of Oka's fundamental papers.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.G. Krantz,   "Function theory of several complex variables" , Wiley  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Oka,   "Collected papers" , Springer  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.M. Range,   "Holomorphic functions and integral representation in several complex variables" , Springer  (1986)  pp. Chapt. 1, Sect. 3</TD></TR></table>