具有有理连通纤维的光滑射影态射的解析与代数 Oka-1 逼近

中文摘要 AI 辅助

设 $\pi:Z\rightarrow Y$ 是复流形的光滑射影态射，其纤维为连通的有理连通纤维。我们证明了在任意紧集上的全纯逼近，以及在任意闭离散集上的有限阶喷射插值，这些结果针对定义在开黎曼曲面上且在那些集合附近全纯的连续提升。对于光滑复代数簇的光滑射影态射以及来自光滑仿射曲线的代数基映射，逼近提升可以选择为代数的，并在任意有限集上具有插值性。在两种情形中，所得提升通过固定基映射的连续提升同伦于初始提升。对于连通的光滑射影复流形，这给出了代数 Oka-1 性质与有理连通性之间的等价性。每个有理连通的光滑射影复流形也是 Oka-1 的。

英文摘要

Let $π:Z\rightarrow Y$ be a smooth projective morphism of complex manifolds with connected rationally connected fibers. We prove holomorphic approximation on arbitrary compact sets and finite-jet interpolation on arbitrary closed discrete sets for continuous liftings defined on open Riemann surfaces and holomorphic near those sets. For smooth projective morphisms of smooth complex algebraic varieties and algebraic base maps from smooth affine curves, the approximating liftings can be chosen algebraic, with interpolation on any finite set. In both cases the resulting lifting is homotopic to the initial one through continuous liftings of the fixed base map. For connected smooth projective complex manifolds, this gives the equivalence between the algebraic Oka-1 property and rational connectedness. Every rationally connected smooth projective complex manifold is also Oka-1.

发表机构

• Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
• State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学重点实验室）
• School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）

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