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

Anar Akhmedov

http://orcid.org/0000-0001-5724-7555

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

E-mail Address: akhmedov@math.umn.edu

Corresponding author.

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and

Sümeyra Sakallı

Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany

E-mail Address: sakal008@math.umn.edu

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https://doi.org/10.1142/S0129167X19500174Cited by:3 (Source: Crossref)

Abstract

We introduce the $2$ $2$ -nodal spherical deformation of certain singular fibers of genus two fibrations, and use such deformations to construct various examples of simply connected minimal symplectic $4$ $4$ -manifolds with small topology. More specifically, we construct new exotic minimal symplectic $4$ $4$ -manifolds homeomorphic but not diffeomorphic to ${ℂ ℙ}^{2} # 6 {\bar{ℂ ℙ}}^{2}$ , ${ℂ ℙ}^{2} # 7 {\bar{ℂ ℙ}}^{2}$ , and $3 {ℂ ℙ}^{2} # k {\bar{ℂ ℙ}}^{2}$ 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 $g \geq 3$ fibrations.

Keywords:

AMSC: 57R55, 57R57, 57M05

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