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Deformation of singular fibers of genus two fibrations and small exotic symplectic 4-manifolds

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Sakallı,  Sümeyra
Max Planck Institute for Mathematics, Max Planck Society;

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1901.11407.pdf
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Citation

Akhmedov, A., & Sakallı, S. (2019). Deformation of singular fibers of genus two fibrations and small exotic symplectic 4-manifolds. International Journal of Mathematics, 30(3): 1950017. doi:10.1142/S0129167X19500174.


Cite as: http://hdl.handle.net/21.11116/0000-0004-6C5E-B
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.