Structures of ε-lc Calabi--Yau pairs

We study the structures of ε-lc Calabi--Yau pairs. We show that such pairs possess a finite cover that is isomorphic in codimension 1 with a fiber product of a Calabi--Yau variety and a rationally connected variety. Moreover, the degree of the cover depends only on ε and the dimension. As a corollary, we show that the index of 4-dimensional non-canonical Calabi--Yau variety is bounded. We also prove a boundedness result on polarized ε-lc Calabi--Yau pairs.