Deformation of Singular Fibers of Genus $2$ Fibrations and Small Exotic Symplectic $4$-Manifolds

Abstract

We introduce the $2$-nodal spherical deformation of certain singular fibers of genus $2$ 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 ${\mathbb{CP}}^2\#6({\overline{\mathbb{CP}}}^2)$, ${\mathbb{CP}}^2\#7({\overline{\mathbb{CP}}}^2)$, and $3{\mathbb{CP}}^2\#k({\overline{\mathbb{CP}}}^2)$ 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.

38 pages, 14 figures. arXiv admin note: text overlap with arXiv:math/0507006 by other authors