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<p>In this paper we begin a systematic study of the class of complex manifolds which are universal targets of holomorphic maps from open Riemann surfaces. We call them Oka-1 manifolds, by analogy with Oka manifolds that are universal targets of holomorphic maps from Stein manifolds of arbitrary dimension. We prove that every complex manifold which is dominable at most points by spanning tubes of complex lines in affine spaces is an Oka-1 manifold. In particular, a manifold dominable by <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3833_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {C}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> at most points is an Oka-1 manifold. We provide many examples of Oka-1 manifolds among compact complex surfaces, including all Kummer surfaces and all elliptic K3 surfaces. We show that the class of Oka-1 manifolds is invariant under Oka-1 maps inducing a surjective homomorphism of fundamental groups; this includes holomorphic fibre bundles with connected Oka fibres. In another direction, we prove that every bordered Riemann surface admits a holomorphic map with dense image in any connected complex manifold. The analogous result is shown for holomorphic Legendrian immersions in an arbitrary connected complex contact manifold.</p>

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Oka-1 manifolds

  • 期刊论文

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  • 发表日期：

    2025年8月21日

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  • Antonio Alarcón,
  • Franc Forstnerič

摘要

In this paper we begin a systematic study of the class of complex manifolds which are universal targets of holomorphic maps from open Riemann surfaces. We call them Oka-1 manifolds, by analogy with Oka manifolds that are universal targets of holomorphic maps from Stein manifolds of arbitrary dimension. We prove that every complex manifold which is dominable at most points by spanning tubes of complex lines in affine spaces is an Oka-1 manifold. In particular, a manifold dominable by \(\mathbb {C}^n\) C n at most points is an Oka-1 manifold. We provide many examples of Oka-1 manifolds among compact complex surfaces, including all Kummer surfaces and all elliptic K3 surfaces. We show that the class of Oka-1 manifolds is invariant under Oka-1 maps inducing a surjective homomorphism of fundamental groups; this includes holomorphic fibre bundles with connected Oka fibres. In another direction, we prove that every bordered Riemann surface admits a holomorphic map with dense image in any connected complex manifold. The analogous result is shown for holomorphic Legendrian immersions in an arbitrary connected complex contact manifold.