AI 中文总结
本文推广Farb与Looijenga的工作，定义椭圆纤维化4流形的光滑窄Mordell-Weil群，构造其与H²(M)平凡格正交补的同构，给出其秩的显式公式并证明有理椭圆曲面是两类Mordell-Weil群秩相同的唯一椭圆曲面。
AI 中文摘要
对于椭圆纤维化4流形π: M→B，我们引入了M的光滑映射类群的一个具有明确几何代表元的子群：光滑窄Mordell-Weil群MW₀(π)。利用Kodaira的奇异纤维分类，我们构造了MW₀(π)与H²(M)中平凡格的正交补之间的自然同构，该对应关系与其全纯版本平行。本文推广了Farb和Looijenga的工作，他们定义了光滑Mordell-Weil群并对仅具有节点纤维的球上纤维化计算了该群。我们给出了MW₀(π)秩的显式公式，该公式仅依赖于基曲线的亏格和π的奇异纤维类型。作为推论，我们证明有理椭圆曲面是唯一一类全纯Mordell-Weil群与光滑Mordell-Weil群秩相同的椭圆曲面。
英文摘要
For an elliptically fibered 4-manifold $π\colon M\to B$, we introduce a subgroup of the smooth mapping class group of $M$ with explicit geometric representatives: the smooth narrow Mordell-Weil group $MW_0(π)$. Leveraging Kodaira's classification of singular fibers, we construct a natural isomorphism between $MW_0(π)$ and the orthogonal complement of the trivial lattice in $H^2(M)$. This identification parallels its holomorphic counterpart. This paper generalizes work of Farb and Looijenga, who defined the smooth Mordell-Weil group and computed it for fibrations over a sphere with only nodal fibers. We give an explicit formula for the rank of $MW_0(π)$ depending only on the genus of the base curve and the types of singular fibers of $π$. As a corollary, we show that rational elliptic surfaces are the only elliptic surfaces with holomorphic and smooth Mordell-Weil groups of the same rank.
Comments20 pages, 3 figures
