Absolute neighbourhood retracts and spaces of holomorphic maps from Stein manifolds to Oka manifolds

The basic result of Oka theory, due to Gromov, states that every continuous map $f$ from a Stein manifold $S$ to an elliptic manifold $X$ can be deformed to a holomorphic map. It is natural to ask whether this can be done for all $f$ at once, in a way that depends continuously on $f$ and leaves $f$ fixed if it is holomorphic to begin with. In other words, is \scrO(S,X) a deformation retract of \scrC(S,X)? We prove that it is if $S$ has a strictly plurisubharmonic Morse exhaustion with finitely many critical points; in particular, if $S$ is affine algebraic. The only property of $X$ used in the proof is the parametric Oka property with approximation with respect to finite polyhedra, so our theorem holds under the weaker assumption that $X$ is an Oka manifold. Our main tool, apart from Oka theory itself, is the theory of absolute neighbourhood retracts. We also make use of the mixed model structure on the category of topological spaces.Comment: Version 2: A few very minor improvements to the exposition. Version 3: Another few very minor improvements to the exposition. To appear in Proceedings AM

Mapping Cylinders and the Oka Principle

We apply concepts and tools from abstract homotopy theory to complex analysis and geometry, continuing our development of the idea that the Oka Principle is about fibrancy in suitable model structures. We explicitly factor a holomorphic map between Stein manifolds through mapping cylinders in three different model structures and use these factorizations to prove implications between ostensibly different Oka properties of complex manifolds and holomorphic maps. We show that for Stein manifolds, several Oka properties coincide and are characterized by the geometric condition of ellipticity. Going beyond the Stein case to a study of cofibrant models of arbitrary complex manifolds, using the Jouanolou Trick, we obtain a geometric characterization of an Oka property for a large class of manifolds, extending our result for Stein manifolds. Finally, we prove a converse Oka Principle saying that certain notions of cofibrancy for manifolds are equivalent to being Stein.Comment: New results included in version

Holomorphic functions of slow growth on nested covering spaces of compact manifolds

Let Y be an infinite covering space of a projective manifold M in P^N of dimension n geq 2. Let C be the intersection with M of at most n-1 generic hypersurfaces of degree d in P^N. The preimage X of C in Y is a connected submanifold. Let phi be the smoothed distance from a fixed point in Y in a metric pulled up from M. Let O_phi(X) be the Hilbert space of holomorphic functions f on X such that f^2 e^(-phi) is integrable on X, and define O_phi(Y) similarly. Our main result is that (under more general hypotheses than described here) the restriction O_phi(Y) to O_phi(X) is an isomorphism for d large enough. This yields new examples of Riemann surfaces and domains of holomorphy in C^n with corona. We consider the important special case when Y is the unit ball B in C^n, and show that for d large enough, every bounded holomorphic function on X extends to a unique function in the intersection of all the nontrivial weighted Bergman spaces on B. Finally, assuming that the covering group is arithmetic, we establish three dichotomies concerning the extension of bounded holomorphic and harmonic functions from X to B

Smooth toric varieties are Oka

In this brief note, we show that every smooth toric variety over the field of complex numbers is an Oka manifold.Comment: Version 2: A remark corrected, a reference updated. Version 3: Two references corrected to match the published version of the monograph [2] (previous references were to a preliminary version of [2]

Eight lectures on Oka manifolds

Over the past decade, the class of Oka manifolds has emerged from Gromov's seminal work on the Oka principle. Roughly speaking, Oka manifolds are complex manifolds that are the target of "many" holomorphic maps from affine spaces. They are "dual" to Stein manifolds and "opposite" to Kobayashi-hyperbolic manifolds. The prototypical examples are complex homogeneous spaces, but there are many other examples: there are many ways to construct new Oka manifolds from old. The class of Oka manifolds has good formal properties, partly explained by a close connection with abstract homotopy theory. These notes were prepared for lectures given at the Institute of Mathematics of the Chinese Academy of Sciences in Beijing in May 2014. They are meant to give an accessible introduction, not to all of Oka theory, but more specifically to Oka manifolds, how they arise and what we know about them.Comment: A few improvements in version