Some degenerations of $G_2$ and Calabi–Yau varieties

Abstract

We introduce a variety ${\hat{G}}_2$ 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 $G_2$, the adjoint variety of the simple Lie group ${\mathbb{G}}_2$. It occurs that it is also the image of ${\mathbb{P}}^5$ by a system of quadrics containing a twisted cubic. Degenerations of this twisted cubic to three lines give rise to degenerations of $G_2$ 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 $g_5^1$ appears as linear sections of the variety ${\hat{G}}_2$.