Branched covers and pencils on hyperelliptic Lefschetz fibrations

Abstract

For every fixed ℎ ≥ 1, we construct an infinite family of simply connected symplectic 4-manifolds 𝑋′_𝑔,ℎ[𝑖], for all 𝑔 > ℎ and 0 ≤ 𝑖 < 2𝑝 − 1, where 𝑝 = ⌊ \frac{𝑔 + 1}{ℎ + 1} ⌋. Each manifold 𝑋′_𝑔,ℎ[𝑖] is the total space of a symplectic genus 𝑔 Lefschetz pencil constructed by an explicit monodromy factorization. We then show that each 𝑋′_𝑔,ℎ[𝑖] is diffeomorphic to a complex surface that is a fiber sum formed from two standard examples of hyperelliptic genus ℎ Lefschetz fibrations, here denoted 𝑍_ℎ and 𝐻_ℎ. Consequently, we see that 𝑍_ℎ, 𝐻_ℎ, and all fiber sums of them admit an infinite family of explicitly described Lefschetz pencils, which we observe are different from families formed by the degree doubling procedure.