Calabi-Yau pairs of complexity two

TL;DR

This paper investigates Calabi-Yau pairs of index one and complexity two, providing criteria to determine when they are of cluster type by analyzing del Pezzo fibrations over toric varieties.

Contribution

It develops a new method to identify cluster type Calabi-Yau pairs of complexity two, linking the problem to del Pezzo fibrations over toric varieties.

Findings

Calabi-Yau pairs of complexity two are of cluster type if and only if certain singularity and volume conditions are met.

The approach reduces the classification problem to studying Gorenstein del Pezzo surfaces with specific properties.

Provides a complete characterization for Gorenstein del Pezzo surfaces of Picard rank one regarding their cluster type.

Abstract

A Calabi-Yau pair of index one and complexity zero is toric. Furthermore, a Calabi-Yau pair of index one and complexity one is of cluster type. In this article, we study Calabi-Yau pairs of index one and complexity two. We develop machinery to decide whether a Calabi-Yau of complexity two is of cluster type. This approach reduces the problem to studying del Pezzo fibrations over toric varieties. We apply this to the setting of Gorenstein del Pezzo surfaces of Picard rank one. We prove that such a surface $X$ is cluster type if and only if $X$ has only $A$-type singularities and either $\mathrm{vol}(X)>1$ or $|X^{\rm sing}|\leq 3$.