Genus 2 Lefschetz fibrations with b 2 + = 1 and c 1 2 = 1 , 2

Abstract

In this article, we construct a family of genus $2$ Lefschetz fibrations $f_n:X_{{\theta }_n}\to {\mathbb{S}}^2$ with $e\left(X_{{\theta }_n}\right)=11$, $b_2^{+}\left(X_{{\theta }_n}\right)=1$, and $c_1^2\left(X_{{\theta }_n}\right)=1$ by applying a single lantern substitution to the twisted fiber sums of Matsumoto’s genus $2$ Lefschetz fibration over ${\mathbb{S}}^2$. Moreover, we compute the fundamental group of $X_{{\theta }_n}$ and show that it is isomorphic to the trivial group if $n=-3$ or $-1$, $\mathbb{Z}$ if $n=-2$, and ${\mathbb{Z}}_{|n+2|}$ for all integers $n\ne -3,-2,-1$. Also, we prove that our fibrations admit $-2$ section, that their total spaces are symplectically minimal, and that they have symplectic Kodaira dimension $\kappa =2$. In addition, using techniques developed over the past decade with other authors, we also construct the genus $2$ Lefschetz fibrations over ${\mathbb{S}}^2$ with $c_1^2=1,2$ and $\chi =1$ via the fiber sums of Matsumoto’s and Xiao’s genus $2$ Lefschetz fibrations, and present some applications in constructing exotic smooth structures on small $4$-manifolds with $b_2^{+}=1$ and $b_2^{+}=3$.