# The de Rham complex of the octonionic projective spaces

The complex projective space $$\mathbb{CP}^n$$ is a complex manifold, and hence its de Rham complex carries a representation of the complex numbers in the form of its complex structure. The quaternionic projective space $$\mathbb{HP}^n$$ is a quaternionic Kähler manifold, and so its de Rham complex carries a local representation of the quaternions. Continuing the analogy, do the octonionic line $$\mathbb{OP}^1$$ and the octonionic projective plane $$\mathbb{OP}^2$$ carry some representation of the octonions, or at least have some extra structure reflecting their octonionic construction?

The Jordan algebra of 3×3 Hermitian octonionic matrices, with multiplication rule $$x\circ y=(xy+yx)/2$$, generates the octonionic projective plane $$\mathbb{OP}^2$$ if we restrict the matrices $$p$$ to unit trace and idempotent ($$p^2=p$$). Lines through the origin containing $$(x,y)$$ with $$x\neq 0$$ equal $$\{(\alpha(y^{-1}x),\alpha):\alpha\in\mathbb{O}\}$$.