On the Complex Cayley Grassmannian

We define a torus action on the (complex) Cayley Grassmannian $X$. Using this action, we prove that $X$ is a singular variety. We also show that the singular locus is smooth and has the same cohomology ring as that of ${\mathbb{C}\mathbb{P}}^5$. Furthermore, we identify the singular locus with a quotient of $G_2^{\mathbb{C}}$ by a parabolic subgroup.