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

## TL;DR

This paper introduces a new deformation technique for singular fibers in genus 2 fibrations and uses it to construct small exotic symplectic 4-manifolds with specific topological properties, expanding the toolkit for 4-manifold construction.

## Contribution

The paper develops 2-nodal spherical deformations of singular fibers and applies them to build new exotic minimal symplectic 4-manifolds with small topology.

## Key findings

- Constructed exotic symplectic 4-manifolds homeomorphic but not diffeomorphic to known manifolds.
- Demonstrated methods to generate manifolds with specific topological invariants.
- Extended deformation techniques to higher genus fibrations.

## 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 $g \geq 3$ fibrations.