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An Oka Manifold? Finnur Lárusson

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An Oka Manifold? Finnur Lárusson WHATIS... ? an Oka Manifold? Finnur Lárusson The prototypical complex manifold is the com- in general, is more important. A complex manifold plex plane C. In three cases out of four we find X is Kobayashi hyperbolic if there is a metric (a something interesting by considering the class nondegenerate distance function) d on X such that of complex manifolds X with “many” or “few” d(f (z), f (w)) ≤ δ(z; w) for all holomorphic maps holomorphic maps X ! C or C ! X. The trick, of f from the open unit disc D = fz 2 C : jzj < 1g to course, is to come up with a fruitful interpretation X, and all z; w 2 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 Cnf0; 1g 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 geometry have been intensively studied play a fundamental role in higher-dimensional in recent years. Many deep problems remain un- complex analysis and complex geometry, similar solved, some to do with a mysterious connection to affine varieties in algebraic geometry. 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 submanifold 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 sheaf a celebrated theorem of G. Faltings. F on X, the cohomology 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; 1) 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 invariant. 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 map 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 theory. 50 Notices of the AMS Volume 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 jT 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 jK 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 homotopy equivalence. Here, the spaces 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 topology. were extended to higher dimensions, 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 complement in Cn of an algebraic or a tame This means that for every x 2 S n K, there is a analytic subvariety of codimension at least 2, the holomorphic function f on S with jf (x)j > sup jf j. K complement in complex projective space 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 torus of dimension at extends to a holomorphic map S ! C. least 2. A Riemann surface 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 vector bundle convex interpolation property (CIP) if, whenever T E over X, such that s(0x) = x and sjEx ! X is a is a contractible subvariety of Cm for some m, every submersion at 0x for all x 2 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 homotopy theory) 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.
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