Non-Collapsible Dual Complexes and Fake del Pezzo Surfaces

TL;DR

This paper introduces a novel method for constructing complex surfaces with specific topological properties using smoothings of normal crossing surfaces with non-collapsible dual complexes, exemplified by the duncehat complex.

Contribution

It presents a new construction technique for complex surfaces with particular invariants, utilizing non-collapsible dual complexes, and provides an explicit example related to the Barlow surface.

Findings

Constructed a complex surface with $h^{1,1} = 9$

Demonstrated the use of non-collapsible dual complexes in surface construction

Proposed a new approach for creating surfaces with $h^{1,0} = h^{2,0} = 0$

Abstract

We propose the new construction of complex surfaces with $h^{1,0} = h^{2,0} = 0$ from smoothings of normal crossing surfaces with non-collapsible dual complexes and carry it out for the simplest case of the duncehat complex, obtaining the surface with $h^{1,1} = 9$ (presumably Barlow surface).