  We develop techniques to construct explicit symplectic Lefschetz fibrations
over the 2-sphere with any prescribed signature and any spin type when the
signature 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 x S^2, with the smallest topology known to date, as well
as larger examples as symplectic Lefschetz fibrations.
