AI 中文总结
研究复射影平面中组合线排列和\((n_k)\) - 构型拓扑与光滑实现的存在性约束，引入奇数和偶数两类特殊线排列，利用相关定理给出约束并建立新下界，表明有限射影平面不存在拓扑实现。
AI 中文摘要
我们研究了复射影平面中组合线排列和\((n_k)\) - 构型的拓扑与光滑实现的存在性约束。通过用局部平坦或光滑嵌入的2 - 球面代替复直线，探讨诸如 Hirzebruch 不等式等经典几何结果在拓扑或光滑范畴中的延续程度。引入了奇数和偶数两类特殊线排列，利用 Furuta 的10/8定理给出了任何光滑实现的非平凡奇数排列的约束，通过研究分支双覆盖并使用G - 符号定理研究拓扑实现的非平凡偶数排列。最后为\((n_k)\) - 构型建立了新的下界\(n \geq k^2 - 5\)，这意味着有限射影平面不存在拓扑实现。
英文摘要
We investigate constraints on the existence of topological and smooth realisations of combinatorial line arrangements and $(n_k)$-configurations in the complex projective plane. By replacing complex lines with locally-flatly or smoothly embedded 2-spheres, we explore the extent to which classical geometric results, such as Hirzebruch's inequality, persist in the topological or smooth category. We introduce two classes of special line arrangements that we call odd and even. We provide constraints for any smoothly realised, non-trivial, odd arrangement via Furuta's 10/8-Theorem. By looking at branched double covers and using the G-signature theorem, we study topologically realised, non-trivial, even arrangement. Finally, we establish a new lower bound for $(n_k)$-configurations, showing that for any topologically realised configuration we have $n \geq k^2-5$, which implies the non-existence of topological realisations for finite projective planes.
Comments13 pages. Comments are welcome
