A complex surface of general type with $p_g=0$, $K^2=2$ and $H_1=\mathbb{Z}/4\mathbb{Z}$

Abstract

We construct a new minimal complex surface of general type with $p_g=0$, $K^2=2$ and $H_1=\mathbb{Z}/4\mathbb{Z}$ (in fact ${\pi }_1^{\text{alg}}=\mathbb{Z}/4\mathbb{Z}$), which settles the existence question for numerical Campedelli surfaces with all possible algebraic fundamental groups. The main techniques involved in the construction are a rational blow-down surgery and a $\mathbb{Q}$-Gorenstein smoothing theory.

24 pages, 25 figures: Simplified the proof of Theorem 5.1; Added Proposition 3.2, Remark 3.3, Remark 6.2 for the ampleness problem; Corrected typos; To appear in Trans. Amer. Math. Soc