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

TL;DR

This paper characterizes the algebraic structure of the compactification of the space of complex associative 3-planes in 7-dimensional complex space, connecting it to octonions and computing its topological invariants.

Contribution

It provides explicit defining equations for the smooth equivariant compactification of the associative Grassmannian in complex 7-space, linking geometric, algebraic, and topological aspects.

Findings

Derived defining equations for the compactification.

Computed the Poincaré polynomial via torus fixed points.

Connected the geometry to quaternionic subalgebras of octonions.

Abstract

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