Oka manifolds

Details Oka manifolds Oka manifolds

2009-06-12

arxiv.org/abs/0906.2421v2

C. R. Math. Acad. Sci. Paris 347 (2009), no. 17-18, 1017-1020

Mathematics

Complex Variables

Dedicated to Mikhael L. Gromov on the occasion of receiving the Abel Prize in May 2009

Scientific paper

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).)

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