The involutive structure on the blow-up of \(\mathbb{R}^n\) in \(\mathbb{C}^n\) (Q1304453)

The blow-up of \(\mathbb{R}^n\) in \(\mathbb{C}^n\) is studied. It is a smooth real \(2n\)-manifold with a distinguished hypersurface. There is a canonical involutive structure on the blow-up (a smooth sub-bundle of the complexified tangent bundle with integrability conditions) given outside the distinguished hypersurface by the complex structure on \(\mathbb{C}^n\setminus \mathbb{R}^n\). It is proved that the involutive structure on the blow-up of \(\mathbb{R}^n\) in \(\mathbb{C}^n\) is hypercomplex. The relationship with the Edge of Wedge Theorem is given.