A log Calabi–Yau pair consists of a proper variety X and a divisor D on it such that KX+ 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 ≤ 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.
|Original language||English (US)|
|Number of pages||31|
|State||Published - Sep 1 2016|
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