The dual complex of Calabi-Yau pairs

Abstract

A log Calabi-Yau pair consists of a proper variety X and a divisor D on it such that is numerically trivial. A folklore conjecture predicts that the dual complex of D is homeomorphic to the quotient of a sphere by a finite group. The main result of the paper shows that the fundamental group of the dual complex of D is a quotient of the fundamental group of the smooth locus of X, hence its pro-finite completion is finite. This leads to a positive answer in dimension 4. We also study the dual complex of degenerations of Calabi-Yau varieties. The key technical result we prove is that, after a volume preserving birational equivalence, the transform of D supports an ample divisor.