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 $q=13,\dots ,19$. 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.

Some revisions are made after the referee's comments. Accepted to Mediterr. J. Math. 24 pages, 10 figures