The dual complex of Calabi-Yau pairs

• 成果类型: Article
• 署名作者: Kollar, Janos; Xu, Chenyang
• 署名单位: Princeton University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-015-0640-6
• 发表日期: 2016
• 页码: 527-557
• 关键词: fundamental-groups; VARIETIES; SINGULARITIES; EXISTENCE

摘要

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.