The coregularity is an invariant that measures a specific type of combinatorial complexity of a pair. We will start this talk by defining this invariant and giving some examples. Then, we will explain how results about complements of Fano varieties of bounded dimension are still valid for Fano varieties of bounded coregularity. Finally, we will show some structural theorems about the orbifold fundamental groups of log Calabi-Yau pairs with bounded coregularity, resembling results for log Fano pairs.

This talk is based on previous works with S. Filipazzi, J. Moraga and J. Peng as well as work in progress with L. Braun.