Pseudoconvex Domains and Oka’s Theorem

Abstract

In this chapter we deal with pseudoconvex domains. In Chap. 4 we saw that the Oka–Cartan Fundamental Theorem holds on holomorphically convex domains, and in Chap. 5 that a holomorphically convex domain is equivalent to a domain of holomorphy. These domains are shown to be pseudoconvex (Cartan–Thullen). The converse (Levi’s problem) was proved by K. Oka and the complete proof is presented here. We will see again that Oka’s Jôku-Ikô plays an essential role.

Historical Supplements

As mentioned at the end of Sect. 7.1, the condition which E.E. Levi [39] himself used to formulate a problem that bears his name is different to that described by plurisubharmonic functions. K. Oka called this problem “Hartogs’ Inverse Problem”; it is reduced to asking that if $- \log \delta _{{{\textrm{P}{\!}\varDelta }}}(x, {\partial }X)$ defined by the boundary distance function $\delta _{{\textrm{P}{\!}\varDelta }}(x, {\partial }X)$ of a domain X, univalent or multivalent, is plurisubharmonic, then X is a domain of holomorphy. Here a function defining the boundary is unnecessary. In this sense, Hartogs’ Inverse Problem has a nuance more general than Levi’s Problem.

Levi’s Problem (Hartogs’ Inverse Problem) was then regarded as the most difficult problem. K. Oka announced the solution of this problem in dimension 2 in 1941 ([63]), and published the full paper in 1942 ([62] VI); then after some interval he proved it for Riemann domains of arbitrary dimension in 1953 ([62] IX); however, he had solved it in a Japanese research report written in 1943.⁶ This fact was written twice at the beginning of the introductions of Oka VIII (1951) and IX (1953), but it has been disregarded historically. In 1954, H.J. Bremermann [7] and F. Norguet [58] gave independently the proofs of Levi’s Problem for univalent domains of general dimension by generalizing Oka’s method (Heftungslemma, Oka VI) to the general dimensional case. As mentioned already, in this problem, even if a domain is given univalently in ${\textbf{C}}^n$, the envelope of holomorphy may be necessarily multivalent over ${\textbf{C}}^n$ (see Example 5.1.5). Therefore, the solution of this problem for univalent domains is not complete. As shown in the proof of this chapter, the difficulty essentially increases from dealing with univalent domains to dealing with multivalent domains (Riemann domains).

It is now made clear by records that the essential part of the solution of Levi’s Problem (Hartogs’ Inverse Problem) in Oka IX is that of his research report written in Japanese, sent to Teiji Takagi (Professor, The Imperial University of Tokyo, well-known as the founder of class field theory) in 1943. During the time Oka wrote and published two papers VII (1950) and VIII (1951), proving Three Coherence Theorems. That purpose was to solve Levi’s Problem (Hartogs’ Inverse Problem) including ramified domains over ${\textbf{C}}^n$ (cf. Introductions of Oka [62] VII, VIII). In the end, Oka restricted himself to the case of unramified domains to write up a paper, and published it (IX); therefore, the Second and Third Coherence Theorems were not used.

As mentioned at the beginning of Sect. 7.4, the proof here given is due to Grauert’s Theorem 7.5.26, which is different to Oka’s original one. As for this proof, H. Grauert wrote in his Collected Volume [26], Vol. I, pp. 155-156 a comment introducing an observation of C.L. Siegel:

Oka’s methods are very complicated. At first he proved (rather simply) that in any unbranched pseudoconvex domain X there is a continuous strictly plurisubharmonic function p(x) which converges to $+\infty$ as x goes to the (ideal) boundary of X. Then he got the existence of holomorphic functions f from this property. In [19] (this is [22] at the end of the present book) the existence of the f comes from a theorem of L. Schwartz in functional analysis (topological vector spaces, see: H. Cartan, Séminaire E.N.S. 1953/54, Exposés XVI and XVII). The approach is much simpler, but my predecessor in Göttingen C.L. Siegel nevertheless did not like it: Oka’s method is constructive and this one is not!

For ramified domains the counter-examples were found later on; in this sense the choice of Oka was right. But it remains in mind that H. Grauert put an emphasis on Levi’s Problem (Hartogs’ Inverse Problem) yet unsolved for ramified domains in his talk at the Memorial Conference of Kiyoshi Oka’s Centennial Birthday on Complex Analysis in Several Variables, Kyoto/Nara 2001.

Exercises

• 1. Show Remark 7.1.7.

• 2. Show Theorem 7.1.42, assuming the continuity of f.

• 3. Show that a complete metric space is Baire.

• 4. Let $\pi :X \rightarrow {\textbf{C}}^n$ be a Riemann domain. Show that there are a Riemannn domain $\pi _0: X_0 \rightarrow {\textbf{C}}^n$ satisfying 7.5.3, and an unramified cover $\lambda : X \rightarrow X_0$ such that $\pi =\pi _0 \circ \lambda$ and $\lambda ^*{\mathscr {O}}(X_0)={\mathscr {O}}(X)$.

• 5. Let $\pi :X \rightarrow {\textbf{C}}^n$ be a Riemann domain. Show that X satisfies 7.5.3 if and only if for any distinct points $a,b \in X$ with $\pi (a)=\pi (b)$ there is an element $f \in {\mathscr {O}}(X)$ with $\underline{f\circ \pi _a^{-1}}_z\not =\underline{f\circ \pi _b^{-1}}_z$, where $\pi _a$ (resp. $\pi _b$) is the local biholomorphism defined by $\pi$ from a neighborhood of a (resp. b) to a neighborhood of z.

• 6. Let $B \subset {\textbf{C}}^n$ be an open ball with center at the origin, and let $X \rightarrow {\textbf{C}}^n$ be a Riemann domain. Define $\delta _B(x, {\partial }X)$ in the same way as $\delta _{{\textrm{P}{\!}\varDelta }}(x, {\partial }X)$ with replacing ${{\textrm{P}{\!}\varDelta }}$ by B.

  Show Lemma 7.5.5 with $\delta _B(z, {\partial }X)$ in place of $\delta _{{\textrm{P}{\!}\varDelta }}(x, {\partial }X)$.

• 7. Show Theorem 7.5.8 with $\delta _B(z, {\partial }X)$ in place of $\delta _{{\textrm{P}{\!}\varDelta }}(x, {\partial }X)$.

• 8. Prove Theorem 7.5.34 (cf. the proof of Theorem 5.4.10).

• 9. Let $\pi :X \rightarrow {\textbf{C}}^n$ be a Riemann domain, and let ${{\textrm{P}{\!}\varDelta }}$ be a polydisk with center at the origin. Let $\tilde{\pi }: \tilde{X} \rightarrow {\textbf{C}}^n$ be a pseudoconvex Riemann domain such that $\tilde{X} \supset X$ and $\tilde{\pi }|_X=\pi$. Assume that for every point $b \in {\partial }X$ (in $\tilde{X}$) there is a neighborhood U of b in $\tilde{X}$, satisfying that $-\log \delta _{{\textrm{P}{\!}\varDelta }}(x, {\partial }X)$ is plurisubharmonic in $x \in U \cap X$. Then, show that X is pseudoconvex. (Note that for a given Riemann domain X there exists always such an $\tilde{X}$ by taking the envelope of holomorphy of X.)

• 10. Let X be a Riemann surface (1-dimensional complex manifold), and let $\varOmega \Subset X$ be a subdomain. Referring to the proofs of Grauert’s Theorem 7.4.1 and Lemma 7.5.28 with $n=1$, show the following:

• a. $\dim _{\textbf{C}}H^1(\varOmega , {\mathscr {O}}_\varOmega ) < \infty$.

• b. Let ${\tilde{\varOmega }}$ be a domain such that $\varOmega \Subset {\tilde{\varOmega }}\Subset X$. Infer from 10a for ${\tilde{\varOmega }}$ that for a point $p \in {\partial }\varOmega$ there is a meromorphic function on ${\tilde{\varOmega }}$ with a pole only at p; show that $\varOmega$ is holomorphically convex.

• c. Show that for a point $q \in \varOmega$ there is a meromorphic function f on ${\tilde{\varOmega }}$ with a pole only at $p_0$ such that $df(q)\not =0$.

• d. Show that for distinct points $q, q' \in \varOmega$ there is a meromorphic function g on ${\tilde{\varOmega }}$ with a pole only at $p_0$ such that $g(q)\not =g(q')$; therefore, $\varOmega$ is Stein. (Thus, we have that $H^1(\varOmega , {\mathscr {O}}_\varOmega )=0$.)

  (Note that every domain of ${\textbf{C}}$ is holomorphically convex.)

• 11. Show that for every domain $\varOmega \subset {\textbf{C}}^n$ there exists a Stein Riemann domain $X \overset{\pi }{{\; \longrightarrow \;}}\ {\textbf{C}}^n$ such that $\varOmega \subset X$ and $\pi |_\varOmega$ is the inclusion map of $\varOmega \subset {\textbf{C}}^n$.