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

TL;DR

This paper constructs a new minimal complex surface of general type with specific invariants, demonstrating the existence of surfaces with particular fundamental groups using advanced surgical and smoothing techniques.

Contribution

It introduces a novel construction of a complex surface with $p_g=0$, $K^2=2$, and fundamental group $bZ/4bZ$, addressing a key existence question in algebraic geometry.

Findings

Existence of a complex surface with specified invariants

Use of rational blow-down surgery and $bQ$-Gorenstein smoothing

Provides new examples for the classification of surfaces

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.