Simply connected symplectic 4-manifolds with \(b_2^+ =1\) and \(c_1^2 =2\)

Abstract

In this article we construct a new family of simply connected symplectic 4-manifolds with \(b_2^+ =1\) and \(c_1^2 =2\) which are not diffeomorphic to rational surfaces by using rational blow-down technique. As a corollary, we conclude that a rational surface \({\mathbf CP}^2 \sharp 7{\bar{{\mathbf CP}}^2}\) admits an exotic smooth structure.