Torsion-dependent Bott–Chern-type decomposition for arbitrary Spin(7)-structures

Construct an operator \(D^{\Phi}_T:C^{\infty}(M)\to\Omega^4_1(M)\oplus\Omega^4_{35}(M)\), together with a suitable first-order differential operator \(d_T\) and harmonic-section space \(H^{\Phi}_T\), such that for an arbitrary Spin(7)-structure with torsion one has \((\Omega^4_1(M)\oplus\Omega^4_{35}(M))\cap\ker d_T=\operatorname{Im}(D^{\Phi}_T)\oplus H^{\Phi}_T\).

Background

The paper proves a Hodge-type decomposition and a global $dd^{\Phi}$-lemma for compact torsion-free Spin(7)-manifolds. These results rely essentially on the preservation of the Spin(7)-type decomposition by the Laplacian, a property that fails in the presence of torsion.

For arbitrary Spin(7)-structures, the paper derives the decomposition of the exterior derivative including torsion terms, but does not obtain an analogue of the torsion-free Bott–Chern-type decomposition. The stated problem asks for a torsion-sensitive Hessian operator, differential complex, and harmonic-section theory that would generalize the torsion-free result.