代数次椭圆性与仿射空间爆破的控制性
Algebraic subellipticity and dominability of blow-ups of affine spaces
摘要：
关于复流形的Oka性质相对于子流形爆破的行为知之甚少。一个流形是$\mathscr A$类的，如果它是一个代数流形中余维数至少为$2$的代数子簇的补，该代数流形是$\mathbb C^n$的Zorski-局部同构。一个$\mathscr A$类的流形是代数次椭圆的，因此是Oka的，而一个$\mathscr A$类的流形在1000个点处被炸毁是$\mathscr A$类的。我们的主要结果是：沿任意代数子流形（不一定连通）爆破的$\mathscr A$类流形是代数次椭圆的。对于代数流形的一般情况下，我们证明了强代数支配性，代数subellipticity的削弱，是由一个任意的爆破保持一个光滑的中心。我们使用的主要结果，以确认预测福斯特的著名猜想，每一个开放的黎曼曲面可以适当的全纯嵌入到$\mathbb C^2$。
英文摘要：
Little is known about the behaviour of the Oka property of a complex manifold with respect to blowing up a submanifold. A manifold is of Class $\mathscr A$ if it is the complement of an algebraic subvariety of codimension at least $2$ in an algebraic manifold that is Zariski-locally isomorphic to $\mathbb C^n$. A manifold of Class $\mathscr A$ is algebraically subelliptic and hence Oka, and a manifold of Class $\mathscr A$ blown up at finitely many points is of Class $\mathscr A$. Our main result is that a manifold of Class $\mathscr A$ blown up along an arbitrary algebraic submanifold (not necessarily connected) is algebraically subelliptic. For algebraic manifolds in general, we prove that strong algebraic dominability, a weakening of algebraic subellipticity, is preserved by an arbitrary blow-up with a smooth centre. We use the main result to confirm a prediction of Forster's famous conjecture that every open Riemann surface may be properly holomorphically embedded into $\mathbb C^2$.
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