Survey of Oka theory

TL;DR

This survey reviews the development of Oka theory in complex geometry, discussing key concepts like Oka manifolds, ellipticity, and their homotopy-theoretic framework, along with recent applications and open problems.

Contribution

It provides a comprehensive overview of Oka theory, including recent advances, geometric conditions, and the integration into homotopy theory, highlighting developments since Gromov's 1989 paper.

Findings

Introduction of Oka manifolds and maps

Ellipticity as a key sufficient condition

Integration of Oka theory into homotopy-theoretic framework

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 first-named author at the conference RAFROT 2010 in Rincon, Puerto Rico, on March 22, 2010, and of a brief survey article by the second-named author in Notices Amer. Math. Soc., January 2010.