Bounded Complements for Fano varieties of coregularity one

The coregularity of a Fano variety is an invariant that is always smaller than the dimension.  One of the qualities of the coregularity is that certain  bounds depending on the dimension extend to bounds depending on the coregularity, hence giving us sharper bounds for high dimension and low coregulairty varieties. By the work of Birkar one can bound the complements of Fano varieties in terms of their dimension. In this talk we will bound the complements of Fano varieties of coregularity one. We also will give a proof of this result for any coregularity under the assumption of  the boundedness of the index and b-representation of Calabi-Yau pairs of fixed dimension. This is joint work with Stefano Filipazzi, Joaquín Moraga and Junyao Peng.