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arXiv · 1706.09952

A counterexample to the birational Torelli problem for Calabi-Yau threefolds

John Christian Ottem·Jørgen Vold Rennemo
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Abstract

The Grassmannian Gr(2,5) is embedded in $\Bbb{P}^9$ via the Plücker embedding. The intersection of two general PGL(10)-translates of Gr(2,5) is a Calabi-Yau 3-fold X, and the intersection of the projective duals of the two translates is another Calabi-Yau 3-fold Y, deformation equivalent to X. Applying results of Kuznetsov and Jiang-Leung-Xie shows that X and Y are derived equivalent, which by a result of Addington implies that their third cohomology groups are isomorphic as polarised Hodge structures. We show that X and Y provide counterexamples to a certain "birational" Torelli statement for Calabi-Yau 3-folds, namely, they are deformation equivalent, derived equivalent, and have isomorphic Hodge structures, but they are not birational.

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John Christian Ottem, Jørgen Vold Rennemo. 2017-10-23. A counterexample to the birational Torelli problem for Calabi-Yau threefolds. https://doi.org/10.1112/jlms.12111

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