The Craighero-Gattazzo surface is simply-connected

TL;DR

This paper proves that the Craighero-Gattazzo surface, a special complex surface with unique properties, is simply-connected, confirming a long-standing conjecture using algebraic and deformation techniques.

Contribution

It provides the first proof of simple connectivity for the Craighero-Gattazzo surface, an explicit example of a complex surface with specific geometric properties.

Findings

The Craighero-Gattazzo surface is simply-connected.

The proof employs algebraic reduction mod p and deformation theory.

This method may be applicable to other complex surface topological questions.

Abstract

We show that the Craighero-Gattazzo surface, the minimal resolution of an explicit complex quintic surface with four elliptic singularities, is simply-connected. This was conjectured by Dolgachev and Werner, who proved that its fundamental group has a trivial profinite completion. The Craighero-Gattazzo surface is the only explicit example of a smooth simply-connected complex surface of geometric genus zero with ample canonical class. We hope that our method will find other applications: to prove a topological fact about a complex surface we use an algebraic reduction mod p technique and deformation theory.