A complex surface of general type with $p_g=0$, $K^2=4$, and $\pi_1=\mathbb{Z}/2\mathbb{Z}$

TL;DR

This paper constructs new complex surfaces and symplectic 4-manifolds with specific invariants using advanced surgical and smoothing techniques, expanding the known examples in complex geometry.

Contribution

It introduces a method to build complex surfaces with $p_g=0$, $K^2=4$, and fundamental group $bZ/2bZ$, and also constructs a related symplectic 4-manifold with different invariants.

Findings

Constructed a minimal complex surface with $p_g=0$, $K^2=4$, $bZ/2bZ$ fundamental group.

Built a symplectic 4-manifold with $b_2^+=1$, $K^2=5$, $bZ/2bZ$ fundamental group.

Abstract

We construct a minimal complex surface of general type with $p_g=0$, $K^2 =4$, and $\pi_1=\mathbb{Z}/2\mathbb{Z}$ using a rational blow-down surgery and a $\mathbb{Q}$-Gorenstein smoothing theory. In a similar fashion, we also construct a symplectic 4-manifold with $b_2^+=1$, $K^2=5$, and $\pi_1=\mathbb{Z}/2\mathbb{Z}$.