Minimality and fiber sum decompositions of Lefschetz fibrations (Q2792150)

Lefschetz fibrations give a way of describing a symplectic 4-manifold in terms of simpler topological data, and have become central in the study of these spaces. The fiber sum operation, an analogue of the standard connected sum which preserves the fibration structure, is one of the main tools of this study, allowing one to realize a topological flexibility in symplectic 4-manifolds not shared by their holomorphic counterparts.NEWLINENEWLINEThe article under review gives a new proof of a result originally due to \textit{M. Usher} [Int. Math. Res. Not. 2006, No. 16, Article ID 49857, 17 p. (2006; Zbl 1110.57017)], that the fiber sum of two nontrivial Lefschetz fibrations over \(S^2\) is minimal; i.e. does not contain an embedded symplectic sphere of square -1. While Usher's original proof relies on relative Gromov invariants and analytic gluing arguments, the argument presented here is substantially shorter, taking advantage of the Lefschetz fibration structure. A further result of Usher, that fiber sums of Lefschetz fibrations produce only symplectic 4-manifolds with positive Kodaira dimension, is obtained as a corollary.NEWLINENEWLINEThe paper also sheds light on the possibilities for fiber-sum decompositions of a given symplectic 4-fold into indecomposable summands, demonstrating that for most pairs \((g,m)\) of positive integers, there exists an infinite family of Lefschetz fibrations of minimal symplectic 4-folds with fiber genus \(g\) and admitting a unique (up to diffeomorphism) decomposition into \(m\) summands (note here that uniqueness is only for the given \(m\); i.e. there may still be decompositions into different numbers of summands). For a given \(g\), the examples are obtained by self-summing \(m\) copies of a certain well-understood (by \textit{A. I. Stipsicz} [Topology Appl. 117, No. 1, 9--21 (2002; Zbl 1007.53059)]) Lefschetz fibration of a ruled surface, in a way which depends on a parameter \(k \in \mathbb{N}\).