Surjective Holomorphic Maps onto Oka Manifolds
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.
KeywordsAlgebraic map Holomorphic map Oka manifold Stein manifold
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.
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