What is an Oka manifold

The prototypical complex manifold is the com-plex plane C. In three cases out of four we find something interesting by considering the class of complex manifolds X with “many ” or “few” holomorphic maps X → C or C → X. The trick, of course, is to come up with a fruitful interpretation of the words “many ” and “few”. As undergraduates, most of us take a course in complex analysis on domains in C. Many of the the-orems proved in such a course extend to a class of manifolds called Stein manifolds. Stein manifolds play a fundamental role in higher-dimensional complex analysis and complex geometry, similar to affine varieties in algebraic geometry. One of the many equivalent definitions of a Stein manifold X says, roughly speaking, that there are many holomorphic maps X → C, enough in fact to embed X as a closed complex submanifold of Cm for some m. Another is the famous Theorem B of H. Cartan that for every coherent analytic sheaf F on X, the cohomology groups Hk(X,F) vanish for all k ≥ 1. A third is a convexity property: there is a proper smooth function X → [0,∞) which is strictly plurisubharmonic. Plurisubharmonicity is ordinary convexity weakened just enough to make it biholomorphically invariant. The equivalence of any two of these definitions is a deep theorem. While it is nontrivial to interpret the word “many”, the word “few ” has a straightforward inter-pretation as “no nonconstant”. A complex manifold X is Brody hyperbolic if every holomorphic map C → X is constant. It turns out that the notion of Kobayashi hyperbolicity, equivalent to Brody hyperbolicity for compact manifolds but stronger Finnur Lárusson is associate professor of mathematics at the University of Adelaide. His email address is finnur