Boundedness of Complements for Log Calabi–Yau Threefolds

Abstract

AbstractIn 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$$\mathcal {N}$$N, such that if a threefold pair$$(X/Z\ni z,B)$$(X/Z∋z,B)has an$$\mathbb {R}$$R-complement which is klt over a neighborhood ofz, then it has ann-complement for some$$n\in \mathcal {N}$$n∈N. We also show the boundedness of complements for$$\mathbb {R}$$R-complementary surface pairs.