Abstract
We investigate the behaviour of the Oka property with respect to deformations of compact complex manifolds. We show that in a family of compact complex manifolds, the set of Oka fibres corresponds to a G δ subset of the base. We give a necessary and sufficient condition for the limit fibre of a sequence of Oka fibres to be Oka in terms of a new uniform Oka property. We show that if the fibres are tori, then the projection is an Oka map. Finally, we consider holomorphic submersions with noncompact fibres.
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References
Birkenhake, C., Lange, H.: Complex tori. In: Progress in Mathematics, vol. 177. Birkhäuser, Basel (1999)
Brody R., Green M.: A family of smooth hyperbolic hypersurfaces in P 3. Duke Math. J. 44, 873–874 (1977)
Forstnerič F.: Oka manifolds. C. R. Acad. Sci. Paris Ser. I 347, 1017–1020 (2009)
Forstnerič, F.: Invariance of the parametric Oka property. In: Complex analysis. Trends in Mathematics, pp. 125–144. Birkhäuser, Basel (2010)
Forstnerič F., Lárusson F.: Survey of Oka theory. N. Y. J. Math. 17a, 11–38 (2011)
Gromov M.: Oka’s principle for holomorphic sections of elliptic bundles. J. Am. Math. Soc. 2, 851–897 (1989)
Kobayashi, S.: Hyperbolic complex spaces. In: Grundlehren der mathematischen Wissenschaften, vol. 318. Springer, Berlin (1998)
Kodaira K., Spencer D.C.: On deformations of complex analytic structures, II. Ann. Math. 67, 403–466 (1958)
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Lárusson, F. Deformations of Oka manifolds. Math. Z. 272, 1051–1058 (2012). https://doi.org/10.1007/s00209-011-0973-9
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DOI: https://doi.org/10.1007/s00209-011-0973-9