A Surface Degeneration With Non-Collapsible Contractible Dual Complex

TL;DR

This paper constructs a specific surface degeneration with a non-collapsible, contractible dual complex, providing new insights into surface degenerations and their topological properties.

Contribution

It introduces a novel example of a surface degeneration with a duncehat dual complex, expanding understanding of surface degenerations with complex topological features.

Findings

Constructed a smoothing of a normal crossing surface with duncehat dual complex

The general fiber has Hodge numbers h^{1,0}=h^{2,0}=0 and h^{1,1}=9

Conjecture that the surface is deformation equivalent to the Barlow surface

Abstract

We construct a smoothing of a particular normal crossing surface, which dual complex is a duncehat, thus obtaining an example of a surface degeneration with non-collapsible, contractible dual complex. The general fiber of this family has $h^{1,0} = h^{2,0} = 0$ and $h^{1,1}=9$, we conjecture that it is in the deformation class of the Barlow surface. This text is a complete rewrite of author's previous work [14].