Indecomposability of certain Lefschetz fibrations (Q2701648)

Let \(f:X\to S^2\) be a Lefschetz fibration which admits a section with self-intersection \(-1\). The main result of this paper states that \(X\) cannot be decomposed as fiber sums, that is, there is no decomposition \(X=X_1 \#_f X_2\) unless \(X_1\) or \(X_2\) is the trivial fibration \(\Sigma_g\times S^2\to S^2\). As a consequence, for any \(g\in\mathbb{N}\), there are infinitely many Lefschetz fibrations of genus \(\geq g\) that are indecomposable (into fiber sums) and do not admit a Kähler structure. In particular, Lefschetz fibrations on symplectic 4-manifolds found in [\textit{S. K. Donaldson}, Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, Vol. II, 309-314 (1998; Zbl 0909.53018)] are indecomposable. This also shows that symplectic Lefschetz fibrations are not necessarily fiber sums of holomorphic ones.