Small Lefschetz fibrations on the BMY line

Abstract

We give explicit positive factorizations for a genus-19 Lefschetz fibration over the torus and a family of genus-4 Lefschetz fibrations over a genus-2 surface. All these examples have only three nodes, and their total spaces are symplectic 4-manifolds of general type on the Bogomolov-Miyaoka-Yau line. In the course of deriving the monodromy relation in genus 19, we obtain several interesting identities, including a factorization of a Dehn twist into a product of three order-3 mapping classes. We show that the total space of this fibration is topologically s-cobordant to the Cartwright-Steger surface, and we determine its real quotient up to homeomorphism. The genus-4 family consists of variations of the Bauer-Catanese fibrations and contains 65 isomorphism classes of Lefschetz fibrations on 45 diffeomorphism types of compact complex ball quotients. We describe their topology and that of the real quotients of the three original surfaces. The handle monodromies in our main factorizations are pseudo-Anosov, and we show more generally that for holomorphic genus-4 Lefschetz fibrations on the BMY line, this holds for every choice of handle generators.