Spin(9) and almost complex structures on 16-dimensional manifolds

TL;DR

This paper explicitly describes the structure of Spin(9) on 16-dimensional manifolds, relating key forms to characteristic polynomials, and derives formulas for Pontrjagin classes and integrals in special holonomy cases.

Contribution

It provides explicit matrix representations of Kähler forms and the 8-form for Spin(9)-structures, linking them to characteristic polynomial coefficients and extending understanding of special holonomy manifolds.

Findings

Explicit formulas for Kähler 2-forms and 8-form Phi

Phi coincides with the fourth coefficient of the characteristic polynomial of psi

Derived formulas for Pontrjagin classes and integrals in Spin(9) holonomy case

Abstract

For a Spin(9)-structure on a Riemannian manifold M^16 we write explicitly the matrix psi of its K\"ahler 2-forms and the canonical 8-form Phi. We then prove that Phi coincides up to a constant with the fourth coefficient of the characteristic polynomial of psi. This is inspired by lower dimensional situations, related to Hopf fibrations and to Spin(7). As applications, formulas are deduced for Pontrjagin classes and integrals of Phi and Phi^2 in the special case of holonomy Spin(9).