Genus two Lefschetz fibrations with $b^{+}_{2}=1$ and ${c_1}^{2}=1,2$

In this article we construct a family of genus two Lefschetz fibrations $f_{n}: X_{\theta_n} \rightarrow \mathbb{S}^{2}$ with $e(X_{\theta_n})=11$, $b^{+}_{2}(X_{\theta_n})=1$, and $c_1^{2}(X_{\theta_n})=1$ by applying a single lantern substitution to the twisted fiber sums of Matsumoto's genus two 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 $n\neq -3, -2, -1$. Also, we prove that our fibrations admit $-2$ section, show that their total space are symplectically minimal, and have the symplectic Kodaira dimension $\kappa = 2$. In addition, using the techniques developed in \cite{A, AP1, ABP, AP2, AZ, AO}, we also construct the genus two 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 two Lefschetz fibrations, and present some applications in constructing exotic smooth structures on small $4$-manifolds with $b^{+}_{2} = 1$ and $b^{+}_{2} = 3$.