Abstract
Let X be a connected Oka manifold, and let S be a Stein manifold with dimS ≥ dimX. We show that every continuous map S → X is homotopic to a surjective strongly dominating holomorphic map S → X. We also find strongly dominating algebraic morphisms from the affine n-space onto any compact n-dimensional algebraically subelliptic manifold. Motivated by these results, we propose a new holomorphic flexibility property of complex manifolds, the basic Oka property with surjectivity, which could potentially provide another characterization of the class of Oka manifolds.
References
R.B. Andrist, N. Shcherbina, E.F. Wold, The Hartogs extension theorem for holomorphic vector bundles and sprays. Ark. Mat. 54(2), 299–319 (2016)
F. Campana, Orbifolds, special varieties and classification theory. Ann. Inst. Fourier (Grenoble) 54(3), 499–630 (2004)
F. Campana, J. Winkelmann, On the h-principle and specialness for complex projective manifolds. Algebr. Geom. 2(3), 298–314 (2015)
B.-Y. Chen, X. Wang, Holomorphic maps with large images. J. Geom. Anal. 25(3), 1520–1546 (2015)
S. Diverio, S. Trapani, Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle (2016). ArXiv e-prints
P.G. Dixon, J. Esterle, Michael’s problem and the Poincaré-Fatou-Bieberbach phenomenon. Bull. Am. Math. Soc. (N.S.) 15(2), 127–187 (1986)
J.E. Fornaess, E.L. Stout, Spreading polydiscs on complex manifolds. Am. J. Math. 99(5), 933–960 (1977)
J.E. Fornaess, E.L. Stout, Regular holomorphic images of balls. Ann. Inst. Fourier (Grenoble) 32(2), 23–36 (1982)
J.E. Fornaess, E.F. Wold, Non-autonomous basins with uniform bounds are elliptic. Proc. Am. Math. Soc. 144(11), 4709–4714 (2016)
F. Forstnerič, The Oka principle for sections of subelliptic submersions. Math. Z. 241(3), 527–551 (2002)
F. Forstnerič, Holomorphic flexibility properties of complex manifolds. Am. J. Math. 128(1), 239–270 (2006)
F. Forstnerič, Runge approximation on convex sets implies the Oka property. Ann. Math. (2) 163(2), 689–707 (2006)
F. Forstnerič, Stein Manifolds and Holomorphic Mappings. The homotopy principle in complex analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 56 (Springer, Heidelberg, 2011).
F. Forstnerič, Oka manifolds: from Oka to Stein and back. Ann. Fac. Sci. Toulouse Math. (6) 22(4), 747–809 (2013). With an appendix by Finnur Lárusson
F. Forstnerič, F. Lárusson, Survey of Oka theory. N. Y. J. Math. 17A, 11–38 (2011)
F. Forstnerič, F. Lárusson, Holomorphic flexibility properties of compact complex surfaces. Int. Math. Res. Not. IMRN 2014(13), 3714–3734 (2014)
M. Gromov, Oka’s principle for holomorphic sections of elliptic bundles. J. Am. Math. Soc. 2(4), 851–897 (1989)
R.C. Gunning, H. Rossi, Analytic Functions of Several Complex Variables (AMS Chelsea Publishing, Providence, RI, 2009). Reprint of the 1965 original
L. Hörmander, An Introduction to Complex Analysis in Several Variables. North-Holland Mathematical Library, vol. 7, 3rd edn. (North-Holland Publishing, Amsterdam, 1990)
S. Kobayashi, T. Ochiai, Meromorphic mappings onto compact complex spaces of general type. Invent. Math. 31(1), 7–16 (1975)
K. Kodaira, Holomorphic mappings of polydiscs into compact complex manifolds. J. Differ. Geom. 6, 33–46 (1971/1972)
F. Kutzschebauch, Flexibility properties in complex analysis and affine algebraic geometry, in Automorphisms in Birational and Affine Geometry. Springer Proceedings in Mathematics & Statistics, vol. 79 (Springer, Cham, 2014), pp. 387–405
F. Larusson, T.T. Truong, Algebraic subellipticity and dominability of blow-ups of affine spaces. Doc. Math. 22, 151–163 (2017)
E. Løw, An explicit holomorphic map of bounded domains in C n with C 2-boundary onto the polydisc. Manuscr. Math. 42(2–3), 105–113 (1983)
R. Nomura, Kähler manifolds with negative holomorphic sectional curvature, Kähler-Ricci flow approach (2016). ArXiv e-prints
V. Tosatti, X. Yang, An extension of a theorem of Wu-Yau (2015). ArXiv e-prints
J. Winkelmann, The Oka-principle for mappings between Riemann surfaces. Enseign. Math. (2) 39(1–2), 143–151 (1993)
D. Wu, S.-T. Yau, A remark on our paper “Negative Holomorphic curvature and positive canonical bundle. Commun. Anal. Geom. 24(4), 901–912 (2016)
D. Wu, S.-T. Yau, Negative holomorphic curvature and positive canonical bundle. Invent. Math. 204(2), 595–604 (2016)
Acknowledgements
The author is supported in part by the grants P3291 and J1-7256 from ARRS, Republic of Slovenia. This work was done during my visit at the Center for Advanced Study in Oslo, and I wish to thank this institution for the invitation, partial support and excellent working condition.
I thank Jörg Winkelmann for having asked the question that is answered (in a more precise form) by Theorems 1.1 and 1.6, and Frédéric Campana for discussions concerning the relationship between the basic Oka property and specialness of compact complex manifolds. These communications took place at the conference Frontiers in Elliptic Holomorphic Geometry in Jevnaker, Norway in October 2016. I thank Finnur Lárusson for helpful suggestions concerning the terminology and the precise statements of Theorems 1.1 and 1.6. Finally, I thank Simone Diverio for references to the recent developments on Kähler manifolds with semi-negative holomorphic sectional curvature, and Tyson Ritter for having pointed out the example by Dixon and Esterle related to Problem 1.5.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG, a part of Springer Nature
About this chapter
Cite this chapter
Forstnerič, F. (2017). Surjective Holomorphic Maps onto Oka Manifolds. In: Angella, D., Medori, C., Tomassini, A. (eds) Complex and Symplectic Geometry. Springer INdAM Series, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-319-62914-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-62914-8_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-62913-1
Online ISBN: 978-3-319-62914-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)