A calibration in $\mathbf R^{16}$ and Federer's product question
Roger Züst
TL;DR
This work constructs a new eight-form calibration $Φ$ on $\mathbf{R}^{16}$ that is self-dual with $Φ^2 = 294\,\mathrm{vol}$ and satisfies a comass bound, enabling a counterexample to Federer’s question: the wedge of two orthogonally supported calibrations need not be a calibration. The authors connect calibrations to spinor theory via Dadok–Harvey’s spinor-product framework, showing $Φ$ arises from the product of two spinors with norms $1$ and $√2$, and they classify the $8$-planes calibrated by $Φ$ into four geometric families (special Lagrangian, complex, Cayley products, and a quaternionic/SU(4)–Sp(4) structured case). Additionally, they derive explicit spinor-product formulas for the calibrated forms $φ_4$ and $φ_8$, expressing them in terms of holomorphic volume forms $Ω,Ω'$, Kähler forms $ω,ω'$, and their real/imaginary parts, and they connect the size of $|Φ^2|$ to systolic-type invariants via the Wirt bound. Overall, the paper extends Cayley-type calibrations to a higher-dimensional, Spin(7)×Spin(7)–type setting and provides a concrete, spinor-based construction of a nontrivial calibration with applications to calibrated geometry and systolic geometry.
Abstract
Building upon the construction of a Cayley calibration adapted to a complex structure, we introduce a calibration $Φ$ in $\bigwedge^8 \mathbf R^{16}$ with $|Φ^2| = 294$. 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 $Φ$ arises from the product of two spinors with norm $1$ and $\sqrt{2}$.