Survey of Oka theory

Abstract

Oka theory has its roots in the classical Oka principle in complex analysis. It has emerged as a subfield of complex geometry in its own right since the appearance of a seminal paper of M. Gromov in 1989. Following a brief review of Stein manifolds, we discuss the recently introduced category of Oka manifolds and Oka maps. We consider geometric sufficient conditions for being Oka, the most important of which is ellipticity, introduced by Gromov. We explain how Oka manifolds and maps naturally fit into an abstract homotopy-theoretic framework. We describe recent applications and some key open problems. This article is a much expanded version of the lecture given by the firstnamed author at the conference RAFROT 2010 in Rincón, Puerto Rico, on 22 March 2010, and of a recent survey article by the second-named author [51].

A Strong Oka Principle for Embeddings of Some Planar Domains into ℂ×ℂ∗

“…In particular, every continuous map from a Stein manifold to an elliptic manifold is homotopic to a holomorphic map. This result, together with various other stronger Oka properties since proved in the literature (for example, see the recent survey [10]), indicate that elliptic manifolds can be thought of as having many maps into them from Stein manifolds. For this reason, elliptic manifolds are suitable targets for acyclic embeddings of Stein manifolds.…”

Monodromy groups of parameterized linear differential equations with regular singularities

“…First note that both V 1 × D and V 2 × D are contractible topological spaces so any vector bundle over these spaces is topologically trivial. Furthermore, both of these sets are Stein manifolds ( [15], p. 209) so any topologically trivial analytic vector bundle is analytically trivial ([14], Satz 2, [12], Corollary 3.2). Therefore, V 1 × D and V 2 × D form a covering such that the vector bundle is analytically trivial on each set.…”