Narrow Results

We introduce the $2$-nodal spherical deformation of certain singular fibers of genus two fibrations, and use such deformations to construct various examples of simply connected minimal symplectic $4$-manifolds with small topology. More specifically, we construct new exotic minimal symplectic $4$-manifolds homeomorphic but not diffeomorphic to ${ℂ\mathbb{P}}^2\#6{\overline{ℂ\mathbb{P}}}^2$, ${ℂ\mathbb{P}}^2\#7{\overline{ℂ\mathbb{P}}}^2$, and $3{ℂ\mathbb{P}}^2\#k{\overline{ℂ\mathbb{P}}}^2$ for $k=16,17,18,19$ using combinations of such deformations, symplectic blowups, and (generalized) rational blowdown surgery. We also discuss generalizing our constructions to higher genus fibrations using $g$-nodal spherical deformations of certain singular fibers of genus $g\ge 3$ fibrations.