Some degenerations of G2G_2 and Calabi–Yau varieties

  • Michał Kapustka

    Jagiellonian University, Krakow, Poland
Some degenerations of $G_2$ and Calabi–Yau varieties cover
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We introduce a variety G^2\hat{G}_2 parameterizing isotropic five-spaces of a general degenerate four-form in a seven dimensional vector space. It is in a natural way a degeneration of the variety G2G_2, the adjoint variety of the simple Lie group G2\mathbb{G}_2. It occurs that it is also the image of P5\mathbb{P}^5 by a system of quadrics containing a twisted cubic. Degenerations of this twisted cubic to three lines give rise to degenerations of G2G_2 which are toric Gorenstein Fano fivefolds. We use these two degenerations to construct geometric transitions between Calabi–Yau threefolds. We prove moreover that every polarized K3 surface of Picard number 2, genus 10, and admitting a g51g^1_5 appears as linear sections of the variety G^2\hat{G}_2.