A gap theorem for minimal log discrepancies of noncanonical singularities in dimension three

We show that there exists a positive real number δ > 0 \delta >0 such that for any normal quasi-projective Q \mathbb {Q} -Gorenstein 3 3 -fold X X , if X X has worse than canonical singularities, that is, the minimal log discrepancy of X X is less than 1 1 , then the minimal log discrepancy of X X is not greater than 1 − δ 1-\delta . As applications, we show that the set of all noncanonical klt Calabi–Yau 3 3 -folds are bounded modulo flops, and the global indices of all klt Calabi–Yau 3 3 -folds are bounded from above.