A simply connected surface of general type with $p_g=0$ and $K^2=3$

Abstract

Motivated by a recent result of Y Lee and the second author [Invent. Math. 170 (2007) 483-505], we construct a simply connected minimal complex surface of general type with $p_g=0$ and $K^2=3$ using a rational blow-down surgery and $\mathbb{Q}$–Gorenstein smoothing theory. In a similar fashion, we also construct a new simply connected symplectic $4$–manifold with $b_2^{+}=1$ and $K^2=4$.