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arXiv · 1707.00534

Intersections of two Grassmannians in $\mathbf{P}^9$

Lev A. Borisov·Andrei Caldararu·Alexander Perry
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Abstract

We study the intersection of two copies of $\mathrm{Gr}(2,5)$ embedded in $\mathbf{P}^9$, and the intersection of the two projectively dual Grassmannians in the dual projective space. These intersections are deformation equivalent, derived equivalent Calabi-Yau threefolds. We prove that generically they are not birational. As a consequence, we obtain a counterexample to the birational Torelli problem for Calabi-Yau threefolds. We also show that these threefolds give a new pair of varieties whose classes in the Grothendieck ring of varieties are not equal, but whose difference is annihilated by a power of the class of the affine line. Our proof of non-birationality involves a detailed study of the moduli stack of Calabi-Yau threefolds of the above type, which may be of independent interest.

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Lev A. Borisov, Andrei Caldararu, Alexander Perry. 2018-04-11. Intersections of two Grassmannians in $\mathbf{P}^9$. https://arxiv.org/abs/1707.00534

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