The dual complex of Calabi--Yau pairs

TL;DR

This paper investigates the topological properties of the dual complex associated with log Calabi--Yau pairs, establishing finiteness results and supporting a conjecture about their homeomorphism type, especially in low dimensions.

Contribution

It proves that the fundamental group of the dual complex is a quotient of the smooth locus's fundamental group, confirming the conjecture in dimensions up to four.

Findings

The fundamental group of the dual complex is finite in certain cases.

The dual complex's fundamental group is a quotient of the smooth locus's fundamental group.

Supports the conjecture that the dual complex is homeomorphic to a sphere quotient.

Abstract

A log Calabi--Yau pair consists of a proper variety $X$ and a divisor $D$ on it such that $K_X+D$ is numerically trivial. A folklore conjecture predicts that the dual complex of $D$ is homeomorphic to the quotient of a sphere by a finite group. The main result of the paper shows that the fundamental group of the dual complex of $D$ is a quotient of the fundamental group of the smooth locus of $X$, hence its pro-finite completion is finite. This leads to a positive answer in dimension $\leq 4$. We also study the dual complex of degenerations of Calabi--Yau varieties. The key technical result we prove is that, after a volume preserving birational equivalence, the transform of $D$ supports an ample divisor.