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

Rampazzo, Marco

doi:10.1090/proc/16745

New counterexamples to the birational Torelli theorem for Calabi–Yau manifolds
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by Marco Rampazzo

Proc. Amer. Math. Soc.

DOI: https://doi.org/10.1090/proc/16745

Published electronically: April 16, 2024

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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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