Oka流形无处不在

中文摘要 AI 辅助

我们证明，每个光滑复射影有理连通流形，因此每个光滑复Fano流形，都是Oka流形，这是通过应用Du、Guo、Wang和Xie的解析判据得出的。我们还建立了Schoen构造中光滑Calabi-Yau三维簇的Oka性质：这些三维簇是射影直线上相对极小的有理椭圆曲面（具有截面且奇异值集合不相交）的纤维积。对于$\PP^n$中次数为$d$的光滑超曲面，其余集是Oka的，等价地说是全纯椭圆的，当且仅当$d\le n+1$。一个具有简化正规交叉边界的光滑连通射影簇，如果它有一条非平凡有理曲线，该曲线在边界下的原像至多包含一个点，并且其拉回的对数切丛是丰富的，则其补集是Oka的。主要正结果来自通过有理曲线形变、沿亏格一纤维的全纯作用以及逐次极方程构造的整曲线全纯族。高次情形的障碍源于Carlson和Griffiths。

英文摘要

We prove that every smooth complex projective rationally connected manifold, and hence every smooth complex Fano manifold, is Oka by applying the analytic criteria of Du, Guo, Wang, and Xie. We also establish the Oka property for the smooth Calabi--Yau threefolds in Schoen's construction: fiber products over the projective line of relatively minimal rational elliptic surfaces with sections and disjoint sets of singular values. For a smooth hypersurface of degree $d$ in $\PP^n$, its complement is Oka, equivalently holomorphically elliptic, if and only if $d\le n+1$. A connected smooth projective variety with reduced simple normal crossings boundary has Oka complement if it admits a nonconstant rational curve whose inverse image of the boundary consists of at most one point and whose pulled-back logarithmic tangent bundle is ample. The main positive results follow from holomorphic families of entire curves constructed by deformations of rational curves, holomorphic actions along genus-one fibers, and successive polar equations. The obstruction in higher degree is due to Carlson and Griffiths.

发表机构

• University of Chinese Academy of Sciences（中国科学院大学）
• 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）

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