Simple normal crossing varieties with prescribed dual complex

TL;DR

This paper demonstrates that any simplicial complex can be realized as the dual complex of a simple normal crossing divisor in a smooth variety, extending previous results on singularities with prescribed dual complexes.

Contribution

It proves the universality of dual complexes for simple normal crossing divisors and simplifies existing constructions for singularities with given dual complexes.

Findings

Any simplicial complex can be realized as a dual complex in a smooth variety.

Extended and simplified the construction of singularities with prescribed dual complexes.

Generalized previous results by Kapovich--Kollar on dual complexes and singularities.

Abstract

We prove that every simplicial complex is the dual complex of some simple normal crossing divisor in a smooth variety. As an application, we simplify and extend the results of Kapovich--Koll\'ar (math.AG:1109.4047) on the existence of singularities with given dual complex.