Mathematical Research Letters

Volume 28 (2021)

Number 6

The double Cayley Grassmannian

Pages: 1765 – 1792


Laurent Manivel (Institut de Mathématiques de Toulouse, Université de Toulouse, France)


We study the smooth projective symmetric variety of Picard number one that compactifies the exceptional complex Lie group $G_2$, by describing it in terms of vector bundles on the spinor variety of $\mathit{Spin}_{14}$. We call it the double Cayley Grassmannian because quite remarkably, it exhibits very similar properties to those of the Cayley Grassmannian (the other symmetric variety of type $G_2$), but doubled in a certain sense. We deduce among other things that all smooth projective symmetric varieties of Picard number one are infinitesimally rigid

Received 14 August 2020

Accepted 10 June 2021

Published 29 August 2022