The dual complex of Calabi–Yau pairs

  title={The dual complex of Calabi–Yau pairs},
  author={J{\'a}nos Koll{\'a}r and Xu Chen},
  journal={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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