On the holomorphicity of genus two Lefschetz fibrations

Abstract

We prove that any genus-2 Lefschetz fibration without reducible fibers and with “transitive monodromy” is holomorphic. The latter condition comprises all cases where the number of singular fibers $\mu\in 10\mathbb{N}$ is not congruent to $0$ modulo $40$. This proves a conjecture of the authors in [SiTi1]. An auxiliary statement of independent interest is the holomorphicity of symplectic surfaces in $S^2$-bundles over $S^2$, of relative degree $\le 7$ over the base, and of symplectic surfaces in $\mathbb{CP}^2$ of degree $\le 17$.