The dual complex of singularities

TL;DR

This paper investigates the dual complex associated with singularities, establishing its properties, minimal representatives, and demonstrating contractibility in specific cases, thereby advancing understanding of singularity topology.

Contribution

It introduces a well-defined minimal dual complex for isolated singularities and relates dual complexes of dlt pairs, showing contractibility in key cases.

Findings

Dual complex of a log terminal singularity is contractible.

Dual complex of a simple normal crossing degeneration is contractible.

A minimal representative of the dual complex is well-defined up to PL homeomorphism.

Abstract

The dual complex of a singularity is defined, up-to homotopy, using resolutions of singularities. In many cases, for instance for isolated singularities, we identify and study a "minimal" representative of the homotopy class that is well defined up-to piecewise linear homeomorphism. This is derived from a more global result concerning dual complexes of dlt pairs. As an application, we also show that the dual complex of a log terminal singularity as well as the one of a simple normal crossing degeneration of a family of rationally connected manifolds are contractible.