AI 中文总结
本文研究同胚但不同胚于 CP^2 的辛 4-流形及其有限阶辛同胚的存在性，将其归结为 12 个接触管道流形的辛填充问题，并确定了伪自由辛 Z_p-作用的阶与局部表示，提出潜在构造或不存在性证明。
AI 中文摘要
本文主要关注一个辛 $4$-流形的存在性，该流形与 $CP^2$ 同胚但不同胚，且允许一个非平凡的有限阶辛同胚。我们将这种 exotic $CP^2$ 的存在性归结为接触几何中的一个问题。更具体地说，我们证明了恰好有 $12$ 个管道流形，每个都配备了与管道相关的典范接触结构，使得存在一个具有非平凡有限阶辛同胚的 exotic $CP^2$ 当且仅当这 $12$ 个接触管道流形中的一个允许某种特定的辛填充。作为主要技术结果，我们完全确定了对于奇数素数 $p$，在具有正典范类的有理同调 $CP^2$ 上的伪自由辛 $Z_p$-作用的阶和局部表示。我们的结果表明，可能存在不是伪造射影平面的有限自同构的辛 $Z_p$-作用。此外，我们引入了一种构造，在关于这 $12$ 个接触管道流形的辛填充的最小假设下，通过逆向工程过程在 Bogomolov-Miyaoka-Yau 线上产生一个辛 $4$-流形。
英文摘要
This paper is primarily concerned with the existence of a symplectic $4$-manifold, homeomorphic but not diffeomorphic to $CP^2$, which admits a nontrivial finite order symplectomorphism. We reduced the existence of such an exotic $CP^2$ to a question in contact geometry. More concretely, we showed that there are precisely $12$ plumbed manifolds, each equipped with a canonical contact structure associated to the plumbing, such that an exotic $CP^2$ with a nontrivial finite order symplectomorphism exists if and only if one of the $12$ contact plumbed manifolds admits a certain specific symplectic filling. As the main technical result, we completely determined the order and local representations of a pseudofree symplectic $Z_p$-action for an odd prime $p$ on a rational homology $CP^2$ with positive canonical class. Our result suggests that potentially, there are symplectic $Z_p$-actions which are not finite automorphisms of a fake projective plane. Moreover, we introduced a construction, which, under the minimal assumptions on the symplectic fillings of the $12$ contact plumbed manifolds, produces a symplectic $4$-manifold on the Bogomolov-Miyaoka-Yau line via a reverse-engineering process.
Comments65 pages, 2 figures, 5 tables
