接触边界无法决定的Stein结构
Stein structures not determined by the contact boundary
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中文总结 AI 辅助
该研究构造了接触边界与第一陈类一致但辛形式不同构的Stein结构，解决了K3问题列表中相对填充问题的归一化版本。
中文摘要 AI 辅助
我们在同一个紧致光滑四维流形上构造了两个Stein结构，它们的边界接触形式被严格等同，且第一陈类一致，但它们的正合辛形式不是辛同构的。此外，没有任何微分同胚能使相关的Weinstein结构同伦。这些例子是通过从一个假射影平面中移除一条光滑双典范曲线的管状邻域，并将得到的Stein结构与其共轭进行比较而得到的。它们的典范Spinc结构相差一个非零的二阶类。在使用公共Boothby-Wang联络对边界形式进行归一化后，一个假设的辛同构会保持定向圆纤维并延拓到除子帽上，这与典范类刚性相矛盾。对于Weinstein的结论，所需的纤维保持性是通过单极Floer分次不对称性获得的，使用了Nelson-Weiler的计算以及Taubes的ECH-Seiberg-Witten对应。这为K3问题列表中问题4.96的相对填充问题的归一化版本提供了肯定的解决方案。
英文摘要
We construct two Stein structures on the same compact smooth four-manifold whose boundary contact forms are strictly identified and whose first Chern classes agree, but whose exact symplectic forms are not symplectomorphic. Moreover, no diffeomorphism makes the associated Weinstein structures homotopic. The examples are obtained by removing a tubular neighborhood of a smooth bicanonical curve from a fake projective plane and comparing the induced Stein structure with its conjugate. Their canonical Spinc structures differ by a nonzero class of order two. After the boundary forms are normalized using a common Boothby-Wang connection form, a hypothetical symplectomorphism preserves the oriented circle fiber and extends over the divisor cap, contradicting canonical-class rigidity. For the Weinstein statement, the required fiber-preservation is obtained from a monopole Floer grading asymmetry, using the Nelson-Weiler computation and Taubes's ECH-Seiberg-Witten correspondence. This gives an affirmative solution to a normalized version of the relative filling problem following Problem 4.96 in the K3 problem list.
