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

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.

30 pages, minor changes

References in corpus (6)

• Homological projective duality for Grassmannians of lines
• Cremona transformations and derived equivalences of K3 surfaces
• Geometric transitions between Calabi-Yau threefolds related to Kustin-Miller unprojections
• Derived equivalence and Grothendieck ring of varieties: the case of K3 surfaces of degree 12 and abelian varieties
• The class of the affine line is a zero divisor in the Grothendieck ring: via $G_2$-Grassmannians
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Cited by in corpus (5)

• Derived equivalence and Grothendieck ring of varieties: the case of K3 surfaces of degree 12 and abelian varieties
• Derived equivalent Hilbert schemes of points on K3 surfaces which are not birational
• An example of birationally inequivalent projective symplectic varieties which are D-equivalent and L-equivalent
• Topics on the geometry of homogeneous spaces
• Calabi-Yau fibrations, simple K-equivalence and mutations