Oka manifolds

TL;DR

This paper proves that various Oka properties of complex manifolds are equivalent, confirming a conjecture related to Gromov's question and establishing that holomorphic maps can be approximated by entire maps under certain conditions.

Contribution

It provides a positive answer to Gromov's question, showing the equivalence of all Oka type properties for complex manifolds, and links approximation of holomorphic maps to the parametric Oka property.

Findings

All Oka properties are equivalent for complex manifolds.

Holomorphic maps from Stein spaces can be approximated by entire maps.

Inclusion of holomorphic maps into continuous maps is a weak homotopy equivalence.

Abstract

We give the following positive answer to Gromov's question (in "Oka's principle for holomorphic sections of elliptic bundles", J. Amer. Math. Soc. 2, 851-897 (1989), 3.4.(D), page 881). THEOREM: If every holomorphic map from a compact convex set in a complex Euclidean space C^n to a certain complex manifold Y is a uniform limit of entire maps of C^n to Y, then Y enjoys the parametric Oka property. In particular, for any reduced Stein space X the inclusion of the space of holomorphic maps of X to Y into the space of continuous maps is a weak homotopy equivalence. This shows that all Oka type properties of a complex manifold are equivalent to each other. (See also the articles F. Forstneric, "Runge approximation on convex sets implies Oka's property", Ann. Math. (2), 163, 689-707 (2006); "Extending holomorphic mappings from subvarieties in Stein manifolds", Ann. Inst. Fourier 55, 733-751 (2005).)