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A counterexample to the birational Torelli problem for Calabi-Yau 3-folds

Ottem, John Christian; Rennemo, Jørgen Vold
Journal article; SubmittedVersion
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Year
2017
Permanent link
http://urn.nb.no/URN:NBN:no-61216

CRIStin
1493903

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  • Matematisk institutt [2137]
  • CRIStin høstingsarkiv [8384]
Original version
arXiv.org. 2017
Abstract
The Grassmannian Gr(2,5) is embedded in ℙ9 via the Pl\"ucker 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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