The smooth Mordell-Weil group and mapping class groups of elliptic surfaces

Abstract

This is a paper in smooth $4$-manifold topology, inspired by the Néron-Lang Theorem in number theory. More precisely, we prove that a smooth version $\MW(π)$ of Mordell-Weil group of an elliptic fibration $π:M\to\Pb^1$ is finitely generated. We compute $\MW(π_d)$ explicitly for elliptic fibrations $π_d:M_d\to\Pb^1$, where $M_d$ is a simply-connected complex surfaces $M_d$ of arithmetic genus $d\ge 1$ and all fibers of ${\pi }_d$ are nodal. We prove in this case that the fibered structure is unique up topological isotopy. By combining this with a result of Donaldson, we obtain the following remarkable consequence: any diffeomorphism of $M_d$ with $d\ge 3$ is topologically isotopic to a diffeomorphism taking fibers to fibers.

32 pages, 4 figures. Final version, to appear in Algebraic Geometry