A simply connected numerical Campedelli surface with an involution

TL;DR

This paper constructs a new simply connected complex surface of general type with specific invariants, using a novel combination of double covering and deformation techniques, and develops methods to analyze its deformation space.

Contribution

It introduces a method for proving unobstructedness of deformations of singular surfaces and constructs a new example of a simply connected numerical Campedelli surface with an involution.

Findings

Constructed a simply connected minimal complex surface with p_g=0 and K^2=2.

Developed a generalized method for proving unobstructedness of deformations.

Analyzed the deformation space and invariant parts under involution.

Abstract

We construct a simply connected minimal complex surface of general type with $p_g=0$ and $K^2=2$ which has an involution such that the minimal resolution of the quotient by the involution is a simply connected minimal complex surface of general type with $p_g=0$ and $K^2=1$. In order to construct the example, we combine a double covering and $\mathbb{Q}$-Gorenstein deformation. Especially, we develop a method for proving unobstructedness for deformations of a singular surface by generalizing a result of Burns and Wahl which characterizes the space of first order deformations of a singular surface with only rational double points. We describe the stable model in the sense of Koll\'ar and Shepherd-Barron of the singular surfaces used for constructing the example. We count the dimension of the invariant part of the deformation space of the example under the induced $\mathbb{Z}/2\mathbb{Z}$-action.