On structures and discrepancies of klt Calabi--Yau pairs

We study the structures of klt Calabi--Yau pairs. We show that the discrepancies of log centers of all klt Calabi--Yau varieties with fixed dimension are in a finite set. As a corollary, we show that the index of 4-dimensional non-canonical Calabi--Yau variety is bounded.
Comment: 28 pages. Remove Theorem 1.3 and 1.8 in the first version