WHATIS... ? an Oka ? Finnur Lárusson

The prototypical is the com- in general, is more important. A complex manifold plex C. In three cases out of four we find X is Kobayashi hyperbolic if there is a (a something interesting by considering the class nondegenerate ) d on X such that of complex X with “many” or “few” d(f (z), f (w)) ≤ δ(z, w) for all holomorphic holomorphic maps X → C or C → X. The trick, of f from the open unit disc D = {z ∈ C : |z| < 1} to course, is to come up with a fruitful interpretation X, and all z, w ∈ D. Here δ denotes the Poincaré of the words “many” and “few”. distance on D. Picard’s little theorem says that the As undergraduates, most of us take a course in twice-punctured plane C\{0, 1} is Brody hyperbolic; complex analysis on domains in C. Many of the the- it is in fact Kobayashi hyperbolic. orems proved in such a course extend to a class of Hyperbolicity problems in higher-dimensional manifolds called Stein manifolds. Stein manifolds complex have been intensively studied play a fundamental role in higher-dimensional in recent years. Many deep problems remain un- complex analysis and , similar solved, some to do with a mysterious to affine varieties in . One of the many equivalent definitions of a Stein with arithmetic. S. Lang conjectured that a smooth manifold X says, roughly speaking, that there are complex projective variety defined over a number many holomorphic maps X → C, enough in fact to field K is Kobayashi hyperbolic if and only if it has embed X as a closed complex of Cm only finitely many rational points over each finite for some m. Another is the famous Theorem B of extension of K. In the one-dimensional case, this is H. Cartan that for every coherent analytic a celebrated theorem of G. Faltings. F on X, the groups Hk(X, F) vanish It is only recently that a good notion of a for all k ≥ 1. A third is a convexity property: there complex manifold X having “many” holomorphic is a proper smooth function X → [0, ∞) which is maps C → X has emerged. The new notion has its strictly plurisubharmonic. Plurisubharmonicity is origins in a seminal paper of M. Gromov, the 2009 ordinary convexity weakened just enough to make Abel laureate, published in 1989 [2]. Gromov’s it biholomorphically . The equivalence of ideas and results have been developed further over any two of these definitions is a deep theorem. the past ten years, primarily by F. Forstneriˇc,partly While it is nontrivial to interpret the word in joint work with J. Prezelj. Forstneriˇchas proved “many”, the word “few” has a straightforward inter- the equivalence of over a dozen properties, saying, pretation as “no nonconstant”. A complex manifold in one way or another, that a complex manifold X is Brody hyperbolic if every holomorphic is the target of many holomorphic maps from C → C X is constant. It turns out that the notion [1]. He has named such manifolds Oka manifolds, of Kobayashi hyperbolicity, equivalent to Brody after K. Oka, a pioneer in several complex variables. hyperbolicity for compact manifolds but stronger In the remainder of this article, we will motivate Finnur Lárusson is associate professor of mathematics at the definition of an Oka manifold, sketch what the University of Adelaide. His email address is finnur. is known about them, and mention two major [email protected]. applications of the ambient .

50 Notices of the AMS 57, Number 1 (What about the fourth class, of complex man- map f : S → X can be deformed to a holomorphic ifolds with “few” holomorphic maps to C? Even map. If f is already holomorphic on a subvariety if we interpret “few” as “no nonconstant”, this T of S, then the restriction f |T may be kept fixed class seems too big to be of interest. It contains all during the deformation. If f is already holomorphic compact manifolds and a whole lot more.) on a holomorphically convex compact subset K of Runge and Weierstrass. The story begins with S, then the restriction f |K may be kept arbitrarily two well-known theorems of nineteenth-century close to being fixed during the deformation. All complex analysis concerning a domain Ω in C. The this can be done parametrically. If we have a family Runge approximation theorem says that if K is a of maps f , depending continuously on a parameter compact subset of Ω with no holes in Ω, then every in a compact subset P of Rk, then the maps can holomorphic map K → C can be approximated, be deformed with continuous dependence on the uniformly on K, by holomorphic maps Ω → C. (By parameter. If the maps parameterized by a compact a holomorphic map K → C we mean a holomorphic subset of P are already holomorphic on S, then function on some open neighborhood of K.) The they may be kept fixed during the deformation. Weierstrass theorem says that if T is a discrete It follows that the inclusion O(S, X) > C(S, X) subset of Ω, then every map T → C extends to a is a weak equivalence. Here, the holomorphic map Ω → C. O(S, X) of holomorphic maps and C(S, X) of In the formative years of modern complex anal- continuous maps S → X are endowed with the ysis, in the mid-twentieth century, these theorems compact-open . were extended to higher , generalizing Ω Examples. The “classical” examples of Oka to a Stein manifold S. The Oka-Weil approximation manifolds, by renowned work of H. Grauert from theorem replaces the topological condition that K around 1960, are complex Lie groups and their have no holes in S with the subtle, nontopological homogeneous spaces. Among other examples are condition that K be holomorphically convex in S. the in Cn of an algebraic or a tame This means that for every x ∈ S \ K, there is a analytic subvariety of at least 2, the f on S with |f (x)| > sup |f |. K complement in complex projective of a The Cartan extension theorem, on the other hand, subvariety of codimension at least 2, Hopf mani- generalises T to a closed complex subvariety of folds, Hirzebruch surfaces, and the complement S and says that every holomorphic map T → C of a finite set in a complex of at extends to a holomorphic map S → C. least 2. A Riemann is Oka if and only We usually consider these theorems as results if it is not hyperbolic. Our understanding of the about Stein manifolds, and of course they are, but geography of Oka manifolds is poor. For example, we can also view them as expressing properties it is an open problem to determine which compact of the target C. We can then formulate them for a complex surfaces are Oka. general target. To avoid topological obstructions, Gromov’s Oka Principle. The most important which are not relevant here, we restrict ourselves to very special S, K, and T . sufficient condition for the Oka property to hold CAP and CIP. A complex manifold X satisfies is ellipticity, introduced by Gromov in [2]. It is yet the convex approximation property (CAP) if, when- another way to say that a complex manifold X is ever K is a convex compact subset of Cm for the target of many holomorphic maps from C. More some m, every holomorphic map K → X can be precisely, X is elliptic if there is a holomorphic approximated, uniformly on K, by holomorphic map s : E → X, called a dominating spray, defined maps Cm → X. A complex manifold X satisfies the on the total space of a holomorphic convex interpolation property (CIP) if, whenever T E over X, such that s(0x) = x and s|Ex → X is a is a contractible subvariety of Cm for some m, every at 0x for all x ∈ X. The theorem that holomorphic map T → X extends to a holomorphic ellipticity implies the Oka property is one version map Cm → X. of Gromov’s Oka principle. It is rather easy to see that CIP implies CAP. (This A Stein manifold is elliptic if and only if it is Oka. is not to say that the Cartan extension theorem There are no known examples of Oka manifolds implies the Oka-Weil approximation theorem: the that are not elliptic. So why focus on the Oka proof that CIP implies CAP uses the Oka-Weil property rather than ellipticity? One reason is that theorem.) Forstneriˇc’swork contains a difficult, the Oka property has good functorial properties roundabout proof of the converse; no simple proof that we cannot at present prove or disprove for is known. ellipticity. We define a complex manifold to be Oka if it Model categories. There is abstract homotopy satisfies the equivalent properties CAP and CIP. theory lurking in the background. The author has Oka Properties. There are more than a dozen shown that the category of complex manifolds can other so-called Oka properties that are nontrivially be embedded into a model category in the sense of equivalent to CAP and CIP. If S is a Stein manifold D. Quillen (roughly speaking, a category in which and X is an Oka manifold, then every continuous one can do ) in such a way that a

January 2010 Notices of the AMS 51 manifold is cofibrant if and only if it is Stein, and fibrant if and only if it is Oka. Applications. The fact that the complement in Cn, n ≥ 2, of an algebraic subvariety of codi- mension at least 2 is Oka is a crucial ingredient Professor of in the proof of Forster’s by Y. Eliash- berg and Gromov, and by J. Schürmann. For each Mathematics n ≥ 2, Forster’s conjecture identifies the smallest N(n) = n+[n/2]+1 such that every n-dimensional The Department of Mathematics Stein manifold embeds into CN(n). at ETH Zurich (www.math.ethz.ch) B. Ivarsson and F. Kutzschebauch have used Gro- mov’s Oka principle, as developed by Forstneriˇc,to invites applications for a faculty solve the holomorphic Vaserstein problem posed by position in mathematics. Applica- Gromov [3]. They show that the inclusion of the ring tions in the fields of algebra and of holomorphic functions on a contractible Stein topology are particularly welcome. manifold into the ring of continuous functions We are looking for candidates with does not induce an isomorphism of K1-groups, whereas by Grauert’s Oka principle it does induce an outstanding research record and an isomorphism of K0-groups. Here, amusingly, a proven ability to direct research Gromov’s Oka principle reveals a limitation of a work of high quality. Willingness more general Oka principle. to teach at all university levels and to participate in collaborative work References [1] F. Forstneriˇc, Oka manifolds, C. R. Math. Acad. Sci. both within and outside of ETH Paris 347 (2009), 1017–20. Zurich is expected. [2] M. Gromov, Oka’s principle for holomorphic sec- tions of elliptic bundles, J. Amer. Math. Soc. 2 (1989), 851–97. In association with other mem- [3] B. Ivarsson and F. Kutzschebauch, A solution of bers of the Department, the future Gromov’s Vaserstein problem, C. R. Math. Acad. Sci. professor will be responsible for Paris 346 (2008), 1239–43. teaching mathematics courses for students of mathematics, natural sciences and . He or she will be expected to teach under- graduate level courses (German or English) and graduate level courses (English).

Please submit your application to- gether with a curriculum vitae, a list of publications, the names of at least three referees, and a short overview of the research interests to the President of ETH Zurich, Prof. Dr. Ralph Eichler, Raemi- strasse 101, ETH Zurich, 8092 Zurich, Switzerland, (or via e-mail to [email protected]), no later than April 15, 2010. With a view toward increasing the number of female professors, ETH Zurich specifically encourages qualified female candidates to apply.

52 Notices of the AMS Volume 57, Number 1