Genus 2 lefschetz fibrations with b⁺₂ = 1 and c²₁ = 1, 2

In this article, we construct a family of genus 2 Lefschetz fibrations f_n: X_θn → S² with e(X_θn) = 11, b⁺₂ (X_θn) = 1, and c²₁(X_θn) = 1 by applying a single lantern substitution to the twisted fiber sums of Matsumoto's genus 2 Lefschetz fibration over S². Moreover, we compute the fundamental group of X_θn and show that it is isomorphic to the trivial group if n = −3 or −1, Z if n = −2, and Z_|n+2_| for all integers n = −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 κ = 2. In addition, using techniques developed over the past decade with other authors, we also construct the genus 2 Lefschetz fibrations over S² with c²₁ = 1,2 and χ = 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.