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

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 Poincaré polynomial of the compactification.

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• Complex $G_2$-manifolds and Seiberg-Witten Equations