A calibration on 𝐑 16 and Federer’s product question

Abstract Building upon the construction of a Cayley calibration adapted to a complex structure, we introduce a calibration Φ in ⋀ 8 R 16 \bigwedge\nolimits^{8}\mathbf{R}^{16} with | Φ 2 | = 294 \lvert\Phi^{2}\rvert=294 . This enables us to show that the product of two orthogonally supported calibrations is not necessarily a calibration, thereby providing a negative answer to a question posed by Federer. Dadok and Harvey developed a general method for constructing calibrations as outer products of two unit spinors in the Clifford algebra. We show that Φ arises from the product of two spinors with norm 1 and 2 \sqrt{2} .