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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 |
Journal/Proceeding Title: | INVENTIONES MATHEMATICAE |
Version: | Author's manuscript |
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