On the Complex Cayley Grassmannian

TL;DR

This paper investigates the complex Cayley Grassmannian, revealing its singular nature, describing its singular locus, and establishing its cohomological properties using a torus action.

Contribution

It introduces a torus action on the complex Cayley Grassmannian and characterizes its singular locus and cohomology, providing new geometric insights.

Findings

The Cayley Grassmannian is singular.

The singular locus is smooth and cohomologically equivalent to ^5.

The singular locus is identified with a quotient of G_2^C.

Abstract

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 $\mathbb{CP}^5$. Furthermore, we identify the singular locus with a quotient of $G_2^\mathbb{C}$ by a parabolic subgroup.