Boundedness of Complements for Log Calabi–Yau Threefolds

In this paper, we study the theory of complements, introduced by Shokurov, for Calabi–Yau type varieties with the coefficient set [0, 1]. We show that there exists a finite set of positive integers N N , such that if a threefold pair (X/Z z,B) (X/Z∋z,B) has an R R -complement which is klt over a neighborhood of z , then it has an n -complement for some n∈ N n∈N . We also show the boundedness of complements for R R -complementary surface pairs.