Log Calabi-Yau Pairs are a generalization of Calabi-Yau varieties, naturally occurring when considering families or branched covers.
The Complexity of a Calabi-Yau pair measures how far it is from being a toric pair. More concretely,  Brown, McKernan, Svaldi and Zong proved that any Calabi-Yau pair of index one and complexity 0 is a toric pair.
Recent work of Mauri and Moraga has studied its crepant birational analogue, the "birational complexity", which measures how far the pair is from admitting a birational toric model.
In this talk we will extend some of the previously known results for Calabi-Yau pairs of index one to arbitrary index. In particular we completely characterize Calabi-Yau pairs of complexity zero and arbitrary index.
This is based on joint work with Joshua Enwright.