A calibration in ${\mathbf{R}}^{16}$ and Federer's product question

Abstract

Building upon the construction of a Cayley calibration adapted to a complex structure, we introduce a calibration $\hat{I}¦$ in ${\bigwedge }^8{\mathbf{R}}^{16}$ with $|\hat{I}{¦}^2|=294$. 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 $\hat{I}¦$ arises from the product of two spinors with norm $1$ and $\sqrt{2}$.