Surface Bundles Over Surfaces of Small Genus,” Geometry and Topology 6

We construct examples of non-isotrivial algebraic families of smooth complex projective curves over a curve of genus 2. This solves a problem from Kirby’s list of problems in low-dimensional topology. Namely, we show that 2 is the smallest possible base genus that can occur in a 4{manifold of non-zero signature which is an oriented ber bundle over a Riemann surface. A rened version of the problem asks for the minimal base genus for xed signature and ber genus. Our constructions also provide new (asymptotic) upper bounds for these numbers