Oka theorems

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 $D_1\times\ldots\times D_n$, where all domains $D_v\subset\mathbf C$, except for, possibly, one, are simply connected.

2) Oka's theorem on the Levi problem: 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 $\mathcal O^p=\mathcal O\times\ldots\times\mathcal O$ ($p$ 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