We show that if the complement of a Donaldson hypersurface in a closed, integral symplectic manifold has the homology of a subcritical Stein manifold, then the hypersurface is of degree one. In particular, this demonstrates a conjecture by Biran and Cieliebak on subcritical polarisations of symplectic manifolds. Our proof is based on a simple homological argument using ideas of Kulkarni–Wood.

中文翻译：

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辛流形的亚临界极化具有一阶

我们表明，如果在封闭的整体辛流形中的唐纳森超曲面的补码具有亚临界Stein流形的同源性，则超曲面的阶为1。特别是，这证明了Biran和Cieliebak对辛歧管的亚临界极化的猜想。我们的证明是基于简单的同质论证，其中使用了库尔卡尼·伍德的思想。