Complex ${\text{G}}_2$ and Associative Grassmannian

We obtain defining equations of the smooth equivariant compactification of the Grassmannian of the complex associative $3$-planes in ${\text{C}}^7$, which is the parametrizing variety of all quaternionic subalgebras of the algebra of complex octonions $\text{OO}\cong {\text{C}}^8$. By studying the torus fixed points, we compute the Poincar'e polynomial of the compactification.