Torus fibrations on symplectic four-manifolds (Q2719819)

The paper comprises two parts: both are concerned with symplectic geometry of four-manifolds fibred by tori. The first part concerns the topological constraints placed on a closed four-manifold by the existence of an integrable system. Its main result is as follows: a symplectic manifold \((X^4,\omega)\) admits a tame essential connected Lagrangian fibration if and only if \(X\) is diffeomorphic to a \(K3\) surface or to a smooth torus bundle over a torus with \(b_1>2\). NEWLINENEWLINENEWLINEThe proof compares ideas from integrable systems, which the author surveys at some length, with results from gauge theory. The second part seeks to understand a particular class of Lefschetz pencils. The main result here is that if \(X^4\) is the total space of a torus bundle over a torus which admits a section, then \(X\) admits a Lefschetz pencil of genus three curves with four base-points. As a corollary, the author gets some non-trivial results on the structure of the genus three maping class group \(\Gamma_3\). In particular, \(\Gamma_3\) admits infinitely many irreducible and inequivalent positive relations.