A $dd^{\Phi}$-Lemma and Bott--Chern-type Cohomology for Spin(7)-Manifolds

Abstract

We study the properties of the $dd^{\Phi}$-operator on $8$-dimensional Spin(7)-manifolds with torsion-free Spin(7)-structures $\Phi$. These operators were first introduced by Harvey and Lawson (An introduction to potential theory in calibrated geometry. Am.J. Math. 131.4 (2009), arXiv:0710.3920). We prove a Hodge decomposition theorem for the $dd^{\Phi}$-operator and obtain an analogue of the $\partial \bar{\partial}$-lemma in K\"ahler geometry. Using this, we define Bott--Chern-type cohomologies for Spin(7)-manifolds. We relate the Bott-Chern-type cohomology spaces to the moduli space of torsion-free Spin(7)-structures and calibrated geometry of Spin(7)-manifolds. These relations naturally give rise to the notion of Cayley-positive cones. In the course of proving the results, we state and prove various identities for the exterior derivative and its decompositions into irreducible Spin(7)-representations as well as identities for second order derivatives and Laplacians. The identities we prove are for any Spin(7)-structures and the specialized torsion-free ones are Spin(7)-analogoues of K\"ahler identities and Bryant--Harvey's identities in the $\mathrm{G}_2$-case (R. Bryant, Some remarks on $\mathrm{G}_2$-structures, Proceedings of the 11th and 12th G\"okova geometry-topology conference, arXiv:math/0305124) and are results of independent interest.