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Collected Papers

Kiyoshi Oka
Publication Date: 
Number of Pages: 
Springer Collected Works in Mathematics
[Reviewed by
James Parson
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Even the title page of the collected-papers volume under review, a reprint of the original edition from 1984, is surprising: the papers of the Japanese mathematician Kiyoshi Oka within are translated into English (from French) by Raghavan Narasimhan, introduced in German by Reinhold Remmert, and presented with commentary in French by Henri Cartan. What sort of mathematician could inspire such a polyglot production with contributions from three significant figures in complex analysis?

In his introduction, Remmert answers this question:

When kings build, carters have work to do.” Kiyoshi Oka was a king. His kingdom was function theory of several complex variables. He solved problems that were regarded as unassailable; he developed methods whose boldness his fellow mathematicians admired. Oka gave complex analysis new life. His ideas continue to have an effect, developed further by mathematicians who themselves are kings. [My translation from Remmert’s German]

The bulk of this volume is made up of Oka’s remarkably focused central group of ten papers, On Analytic Functions of Several Variables I–X, published between 1936 and 1962. As Oka explains in the introduction to the first paper, his goal in this series is to treat the major open problems discussed in Theorie der Funktionen Mehrerer Komplexer Veränderlichen (1934) by Heinrich Behnke and Peter Thullen. This brief yet influential volume in Springer’s Ergebnisse series (a report rather than a textbook) laid out the state of complex analysis in several variables in 1934 and presented the key challenges for further development of the theory.

For those familiar only with the one-variable theory, the character of complex analysis in several variables is rather surprising. As Oka writes in the introduction to his ninth paper:

The general theory of analytic continuation in a single variable is like an open field; despite many efforts one has not found any facts which could not have been predicted by formal logic. The case of several variables, on the other hand, seems to us like mountainous country, very precipitous.

A key feature of the several-variable theory is the subtle nature of the domains of holomorphic functions. In the one-variable theory, any non-empty open subset of the complex numbers is a domain of holomorphy, in the sense that it admits a holomorphic function that cannot be extended holomorphically to any larger open subset. Furthermore, the Riemann Mapping Theorem says that any two proper simply-connected open subsets of the complex numbers are holomorphically equivalent. Both of these facts in one dimension fail dramatically in two or more dimensions; basic function-theoretic results depend heavily on delicate properties of domains.

Oka’s first paper (1936) provides an example of this importance of domains. One of the fundamental problems presented in Behnke-Thullen is what Cartan later dubbed “the first Cousin problem” after Pierre Cousin. In one variable, one has the Mittag-Leffler theorem (1876): given an open subset \(U\subseteq \mathbf{C}\) and a discrete set of points \(z_i\in U\) together with specifications of desired meromorphic principal parts \(p_i\) at each \(z_i\), there is a meromorphic function on \(U\) whose singularities occur precisely at the \(z_i\) and have precisely the given principal parts \(p_i\). Pierre Cousin (1895) proved an analogous theorem for open sets in \(\mathbf{C}^n\) of the form \(X_1\times\dotsb\times X_n\), where the \(X_i\) are non-empty bounded open subsets of \(\mathbf{C}\). (It is already a tricky problem to decide how to give the data to pose a Mittag-Leffler-type problem; Poincaré (1883) and Cousin found a good way to do this sort of thing.) Oka’s paper extends this result as follows: let \(Y_1,\dotsc,Y_\nu\) be non-empty bounded open subsets of \(\mathbf{C}\), and let \(R_j\) (for \(j=1,\dotsc,\nu\)) be rational functions on \(\mathbf{C}^n\). Let \(\Delta\subset\prod_i X_i\) be defined by \(R_j(x)\in Y_j\) for \(j=1,\dotsc,\nu\). Oka’s main result is that a Mittag-Leffler-type problem (the first Cousin problem) on such a \(\Delta\) can always be solved.

In his second paper (1937), Oka follows the same approach he used in the first paper to prove that the first Cousin problem can always be solved on domains of holomorphy (a type of domain that includes the “rationally convex” domains he considered in his first paper), resolving a major open problem and providing a strong generalization of Mittag-Leffler’s result to higher dimensions. In his third paper (1939), Oka solves the “second Cousin problem” (another major open problem) for domains of holomorphy: just as the first Cousin problem seeks generalizations of the Mittag-Leffler theorem, the second Cousin problem seeks generalizations of Weierstrass’s theorem that constructs holomorphic functions having specified zeros and vanishing orders. Cousin himself studied this problem in product domains \(X_1\times\dotsb\times X_n\) using the same methods that he applied to generalizing the Mittag-Leffler theorem. He made a subtle error in his proof, however, which was only discovered in 1917 by Thomas Grönwall, who discovered that there is a topological obstruction to finding holomorphic functions with specified zeros. Oka’s result in his third paper is that for domains of holomorphy, if the topological obstruction to solving the second Cousin problem vanishes, then it can be solved. Such a result (the vanishing of a topological obstruction implies the possibility of a holomorphic construction) is now generally called an Oka principle, and there has been much further work to prove similar facts in other contexts.

Besides the two Cousin problems, a major topic in Oka’s papers is the so-called Levi problem. This problem asks for a characterization of the sort of special domains that feature in his early papers (domains of holomorphy) by conditions on their boundaries, namely by a “pseudoconvexity” property. In his sixth paper (1942), Oka studies this circle of problems for domains in \(\mathbf{C}^2\), proving such characterizations of domains of holomorphy by boundary conditions. Along the way, he introduces the important notion of pseudo-convex functions. (Pierre Lelong introduced this notion independently (also in 1942), calling such functions plurisubharmonic, which is the name now generally in use.) Oka returns to these topics in his ninth paper (1953), which Cartan describes as “the coronation of Oka’s oeuvre.” In that paper, Oka revisits all of the themes from his earlier work (Cousin problems as well as Levi-type problems), generalizing everything from domains in \(\mathbf{C}^n\) to “unramified domains,” by which Oka means manifolds \(D\) equipped with holomorphic mappings \(D\to\mathbf{C}^n\) that are local homeomorphisms (on \(D\)); in the standard terminology of algebraic geometry, one would say that such a \(D\) is étale over \(\mathbf{C}^n\). (The interest in these unramified domains goes back to Riemann’s original conception of a Riemann surface as a surface spread out in several sheets over a domain in \(\mathbf{C}\).)

While working on these fundamental problems in several complex variables, Oka met numerous examples of what he described as “arithmetical notions.” His remarkable seventh paper (published in 1950) addresses these issues, which have to do with ideals in rings of holomorphic functions. As Oka explains, many important problems naturally require one to consider ideals over variable open sets in a fixed domain \(D\). In the terminology that Cartan imported to complex analysis from Jean Leray’s wartime work on algebraic topology, such structures are sheaves of ideals in the sheaf of rings of holomorphic functions on \(D\). Cartan had begun to formulate a theory of such ideals in a paper published in 1940, but during the war years, he and Oka worked independently on these topics.

Oka’s results include local-to-global results in the same spirit as his results (solution to Cousin problems) from his first three papers discussed above. A central result of a different character, though, in the paper is a basic fact about homogeneous linear equations with holomorphic-function coefficients. In linear algebra, of course, one knows that a homogeneous linear equation has a finite-dimensional space of solutions. In his paper, Oka considers a homogeneous linear equation \(\sum_{j=1}^p A_j F_j = 0\) with coefficients \(F_j\) and unknowns \(A_j\), all taken to be holomorphic functions on a domain in \(\mathbf{C}^n\). He proves a fundamental finiteness result for the space of solutions to such an equation generalizing the basic finiteness result from linear algebra; his result is now known as Oka’s coherence theorem. As he explains in his introduction, Oka took particular pride in this seventh paper, since he viewed the arithmetic problems that he studied as the starting point for a new field of mathematical exploration potentially of the same breadth as the field that the early investigators of functions of several complex variables had opened for him to explore.

Each paper in this volume is accompanied by a commentary (in French) by Henri Cartan, outlining the contents and arguments of the article. The commentaries are quite valuable guides to the articles. Cartan often connects Oka’s arguments and results with later developments, rephrasing some of what Oka did in the language of sheaves and sheaf cohomology that later become standard in the subject. It is quite remarkable that Oka’s great contemporary — and his best reader — should provide such commentary. If only every collected-works volume received this kind of attention!

The papers of Oka are not often read today perhaps in large part because of Henri Cartan. In the remarkable 1951–2 session of the Séminaire Cartan, Cartan together with a group of brilliant collaborators (including, among others, Jean-Pierre Serre) reworked much of the Cartan-Oka theory systematically using the new language of sheaves and sheaf cohomology. Many of Oka’s results could be compressed into a single phrase: the sheaf of holomorphic functions on a complex analytic space is coherent, and coherent sheaves on Stein spaces satisfy Cartan’s Theorem A (global generation) and Theorem B (vanishing of higher cohomology).

After this seminar, the field moved quickly: a new generation took up the language of sheaves and sheaf cohomology, which proved remarkably fruitful through the 1960s. Remmert along with his fellow Behnke student Hans Grauert were chief members of the group applying these new techniques. At the same time, Serre unified the languages of several complex variables and algebraic geometry by rephrasing algebraic geometry using sheaves and sheaf cohomology, proving theorems analogous to those from the 1951–2 seminar in the algebraic context. This approach to algebraic geometry became the foundation for Grothendieck’s theory of schemes, and in the early development of this theory, there was much contact and sharing of ideas between the analytic and algebraic fields.

Despite the fact that one can treat Oka’s results systematically within a vastly general framework, thirty years ago, three of the architects of that framework (Cartan, Remmert, and Narasimham) felt it worthwhile to provide a fine edition of the original works. For a casual reader who has learned the modern approaches, it is rewarding to see the problems solved “with bare hands,” stripped of the later conceptual machinery. This sort of reading helps us understand the role of the new tools: where they smooth awkward arguments, where they organize varied phenomena, and where they may obscure simple ideas. Seeing the richness of particular instances of general results also increases our pleasure in them — a special case is often more intriguing than a general theorem — and it heightens our attention to their proofs.

Although his work had spurred much research and exploration, by the publication of the tenth paper in his great series, Oka seems disappointed with the direction of work in his field and of mathematics in general:

I shall not go into technical detail in this introduction. Rather, I should like to refer to the feeling for the seasons, which has been special to Japanese people since time immemorial, to explain what I feel in completing the present Memoir.
There is a tendency towards abstraction in the progress of the mathematical sciences of today.
Even in the field of our own research, theorems have become more and more general, and some of them have even gone outside the space of complex variables. I felt that this was winter. I have waited for a long time for the return of spring and have wanted to make some studies which would make this felt. The present memoir is but the first result.

Oka died 16 years later, at the age of 76, without publishing any further works.

James Parson is an assistant professor of mathematics at Hood College in Frederick, Maryland.

Rationally Convex Domains

Domains of Holomorphy

The Second Cousin Problem

Domains of Holomorphy and Rationally Convex Domains

The Cauchy Integral

Pseudo convex Domains

On Some Arithmetical Notions

Fundamental Lemma

Unramified Domains Without Points at Infinity

A New Method of Generating Pseudoconvex Domains.