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The dual complex of Calabi-Yau pairs

Author(s): Kollar, Janos; Xu, Chenyang

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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.
Publication Date: Sep-2016
Electronic Publication Date: 14-Dec-2015
Citation: Kollar, Janos, Xu, Chenyang. (2016). The dual complex of Calabi-Yau pairs. INVENTIONES MATHEMATICAE, 205 (527 - 557. doi:10.1007/s00222-015-0640-6
DOI: doi:10.1007/s00222-015-0640-6
ISSN: 0020-9910
EISSN: 1432-1297
Pages: 527 - 557
Type of Material: Journal Article
Version: Author's manuscript

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