Oka-1 manifolds: New examples and properties
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910020625498112 |
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| author | Forstneric, Franc Larusson, Finnur |
| author_facet | Forstneric, Franc Larusson, Finnur |
| contents | In this paper we investigate Oka-1 manifolds and Oka-1 maps, a class of complex manifolds and holomorphic maps recently introduced by Alarcón and Forstnerič. Oka-1 manifolds are characterised by the property that holomorphic maps from any open Riemann surface to the manifold satisfy the Runge approximation and Weierstrass interpolation conditions, while Oka-1 maps enjoy similar properties for liftings of maps from open Riemann surfaces in the absence of topological obstructions. We also formulate and study the algebraic version of the Oka-1 condition, called aOka-1. We show that it is a birational invariant for compact algebraic manifolds and holds for all uniformly rational projective manifolds. This gives a Runge approximation theorem for maps from compact Riemann surfaces to uniformly rational projective manifolds. Finally, we study a class of complex manifolds with an approximation property for holomorphic sprays of discs. This class lies between the smaller class of Oka manifolds and the bigger class of Oka-1 manifolds and has interesting functorial properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09798 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Oka-1 manifolds: New examples and properties Forstneric, Franc Larusson, Finnur Complex Variables Algebraic Geometry Primary 32E30, 32Q56. Secondary 14H55, 14M20, 14M22 In this paper we investigate Oka-1 manifolds and Oka-1 maps, a class of complex manifolds and holomorphic maps recently introduced by Alarcón and Forstnerič. Oka-1 manifolds are characterised by the property that holomorphic maps from any open Riemann surface to the manifold satisfy the Runge approximation and Weierstrass interpolation conditions, while Oka-1 maps enjoy similar properties for liftings of maps from open Riemann surfaces in the absence of topological obstructions. We also formulate and study the algebraic version of the Oka-1 condition, called aOka-1. We show that it is a birational invariant for compact algebraic manifolds and holds for all uniformly rational projective manifolds. This gives a Runge approximation theorem for maps from compact Riemann surfaces to uniformly rational projective manifolds. Finally, we study a class of complex manifolds with an approximation property for holomorphic sprays of discs. This class lies between the smaller class of Oka manifolds and the bigger class of Oka-1 manifolds and has interesting functorial properties. |
| title | Oka-1 manifolds: New examples and properties |
| topic | Complex Variables Algebraic Geometry Primary 32E30, 32Q56. Secondary 14H55, 14M20, 14M22 |
| url | https://arxiv.org/abs/2402.09798 |