A log Calabi–Yau pair consists of a proper variety X and a divisor D on it such that \(K_X+D\) 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 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.
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Many people seem to have been aware of this question, among others M. Gross, S. Keel. V. Shokurov, but we could not find any specific mention in the literature.
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We thank M. Gross, A. Levine, J. Nicaise, S. Payne, N. Sibilla and C. Simpson for comments, discussions, references and J. McKernan for sharing with us early versions of . We also want to thank the anonymous referee for many helpful remarks. Partial financial support to JK was provided by the NSF under grant number DMS-1362960. Partial financial support to CX was provided by The National Science Fund for Distinguished Young Scholars. A large part of this work was done while CX enjoyed the inspiring environment at the Institute for Advanced Studies.
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Kollár, J., Xu, C. The dual complex of Calabi–Yau pairs. Invent. math. 205, 527–557 (2016). https://doi.org/10.1007/s00222-015-0640-6