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The dual complex of Calabi–Yau pairs

Abstract

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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Notes

  1. 1.

    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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Acknowledgments

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 [3]. 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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Correspondence to Chenyang Xu.

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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

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