Lefschetz fibrations with arbitrary signature

Abstract

We develop techniques to construct explicit symplectic Lefschetz fibrations over the $2$-sphere with any prescribed signature $\sigma$ and any spin type when $\sigma$ is divisible by $16$. This solves a long-standing conjecture on the existence of such fibrations with positive signature. As applications, we produce symplectic $4$-manifolds that are homeomorphic but not diffeomorphic to connected sums of $S^2 \times S^2$, with the smallest topology known to date, as well as larger examples as symplectic Lefschetz fibrations.