# The dual complex of Calabi–Yau pairs

@article{Kollr2015TheDC,
title={The dual complex of Calabi–Yau pairs},
author={J{\'a}nos Koll{\'a}r and Xu Chen},
journal={Inventiones mathematicae},
year={2015},
volume={205},
pages={527-557}
}
• Published 28 March 2015
• Mathematics
• Inventiones mathematicae
A log Calabi–Yau pair consists of a proper variety X and a divisor D on it such that $$K_X+D$$KX+D 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 $$\le$$≤4. We…
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