Relations between indices of Calabi--Yau varieties and pairs

We show that for any smooth Calabi–Yau variety, its index can be realized as the index of a Kawamata log terminal (klt) Calabi–Yau pair of lower dimension with standard coefficients. Our approach is based on an inductive argument on the dimension using the Beauville–Bogomolov decomposition. A key step in the argument is to prove that for $n\ge 3$, any positive integer $m$ satisfying $\phi (m)\le 2n$ can be realized as the index of a klt Calabi–Yau pair of dimension $n-1$.