Genus-$3$ Lefschetz Fibrations and Exotic $4$-Manifolds with $b_{2}^{+}=3$

Abstract

We explicitly construct a genus-$3$ Lefschetz fibration over $\mathbb{S}^{2}$ whose total space is $\mathbb{T}^{2}\times \mathbb{S}^{2}\# 6\overline{\mathbb{C} P^{2}}$ using the monodromy of Matsumoto's genus-$2$ Lefschetz fibration. We then construct more genus-$3$ Lefschetz fibrations whose total spaces are exotic minimal symplectic $4$-manifolds $3 \mathbb{C} P^{2} \# q\overline{\mathbb{C} P^{2}}$ for $q=13,\ldots,19$. We also generalize our construction to get genus-$3k$ Lefschetz fibration structure on the $4$-manifold $\Sigma_{k}\times \mathbb{S}^{2}\# 6\overline{\mathbb{C} P^{2}}$ using the generalized Matsumoto's genus-$2k$ Lefschetz fibration. From this generalized version, we derive further exotic $4$-manifolds via Luttinger surgery and twisted fiber sum.