Octonionic Calabi-Yau theorem (Q6596295)

This article concerns octonionic geometry in two octonionic, or 16 real, dimensions. The authors give an in-depth of octonionics and two-dimensional octonionic linear algebra. Then they introduce the notion of an \(\mathrm{GL}_2(\mathbb O)\)-affine structure on a (real) 16-dimensional manifold. They show that \(\mathrm{GL}_2(\mathbb O)\)-affine manifolds admit an analogue of the local \(dd^c\)-lemma, as well as an octonionic analogue of a Kähler metric that is locally given by the Hessian of a smooth potential. Finally, they introduce an octonionic equation of Monge-Ampère type which makes sense on any \(\mathrm{GL}_2(\mathbb O)\)-affine manifold and prove that it can be solved on a subclass of manifolds that they call Spin\((9)\)-affine (beyond tori \(\mathbb{O}^2/\Lambda\), no examples are exhibited). Proving this, their main result takes up the second half of the paper and establishes a result analogous to the Calabi-Yau theorem from complex geometry in this two-dimensional octonionic setting.