Small Lefschetz Fibrations on Simply-Connected $4$-Manifolds

Tulin Altunoz

Published 2021-04-03Version 1

We consider simply-connected $4$-manifolds admitting Lefschetz fibrations over the $2$-sphere. We explicitly construct Lefschetz fibrations of genus $3$ and $4$ on simply-connected $4$-manifolds which are exotic symplectic $4$-manifolds in the homeomorphism classes of $\mathbb{C} P^{2}\#7\overline{\mathbb{C} P^{2}}$ and $\mathbb{C} P^{2}\#8\overline{\mathbb{C} P^{2}}$, respectively. From these, we provide upper bounds for the minimal number of singular fibers of such fibrations. In addition, we prove that this number is equal to $14$ for $g=2$. When such fibrations are hyperelliptic, we prove that it is $18$ for $g=3$. Moreover, we discuss the number for higher genera.