New counterexamples to the birational Torelli theorem for Calabi--Yau manifolds

Marco Rampazzo

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arXiv · 2211.03702

New counterexamples to the birational Torelli theorem for Calabi--Yau manifolds

Marco Rampazzo

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Abstract

We produce counterexamples to the birational Torelli theorem for Calabi-Yau manifolds in arbitrarily high dimension: this is done by exhibiting a series of non birational pairs of Calabi-Yau $(n^2-1)$-folds which, for $n \geq 2$ even, admit an isometry between their middle cohomologies. These varieties also satisfy an $\mathbb L$-equivalence relation in the Grothendieck ring of varieties, i.e. the difference of their classes annihilates a power of the class of the affine line. We state this last property for a broader class of Calabi-Yau pairs, namely all those which are realized as pushforwards of a general $(1,1)$-section on a homogeneous roof in the sense of Kanemitsu, along its two extremal contractions.

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Marco Rampazzo. 2022-11-07. New counterexamples to the birational Torelli theorem for Calabi--Yau manifolds. https://arxiv.org/abs/2211.03702

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