New York Journal of Mathematics
Volume 17a (2011) 11-38


Franc Forstnerič and Finnur Lárusson

Survey of Oka theory

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Published: January 30, 2011
Keywords: Oka principle, Stein manifold, elliptic manifold, Oka manifold, Oka map, subelliptic submersion, model category
Subject: Primary 32E10. Secondary 18G55, 32E30, 32H02, 32Q28, 55U35.

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 Rincón, Puerto Rico, on 22 March 2010, and of a recent survey article by the second-named author, 2010.


The first-named author was partially supported by grants P1-0291 and J1-2043-0101, ARRS, Republic of Slovenia, and by the conference RAFROT, Rincón, Puerto Rico, March 2010.

Author information

Franc Forstnerič:
Franc Forstnerič, Faculty of Mathematics and Physics, University of Ljubljana, and Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia

Finnur Lárusson:
Finnur Lárusson, School of Mathematical Sciences, University of Adelaide, Adelaide SA 5005, Australia