We study the geometric properties of a $(2m+1)$-dimensional complex manifold $\mathcal{ M}$ admitting a holomorphic reduction of the frame bundle to the structure group $P \subset \mathrm{ Spin}(2m+1,\mathbb{ C})$, the stabiliser of the line spanned by a pure spinor at a point. Geometrically, $\mathcal{ M}$ is endowed with a holomorphic metric $g$, a holomorphic volume form, a spin structure compatible with $g$, and a holomorphic pure spinor field $\xi$ up to scale. The defining property of $\xi$ is that it determines an almost null structure, i.e.\ an $m$-plane distribution $\mathcal{ N}_\xi$ along which $g$ is totally degenerate. We develop a spinor calculus, by means of which we encode the geometric properties of $\mathcal{ N}_\xi$ and of its rank-$(m+1)$ orthogonal complement $\mathcal{ N}_\xi^\perp$ corresponding to the algebraic properties of the intrinsic torsion of the $P$-structure. This is the failure of the Levi-Civita connection $\nabla$ of $g$ to be compatible with the $P$-structure. In a similar way, we examine the algebraic properties of the curvature of $\nabla$. Applications to spinorial differential equations are given. Notably, we relate the integrability properties of $\mathcal{ N}_\xi$ and $\mathcal{ N}_\xi^\perp$ to the existence of solutions of odd-dimensional versions of the zero-rest-mass field equation. We give necessary and sufficient conditions for the almost null structure associated to a pure conformal Killing spinor to be integrable. Finally, we conjecture a Goldberg--Sachs-type theorem on the existence of a certain class of almost null structures when $(\mathcal{ M},g)$ has prescribed curvature. We discuss applications of this work to the study of real pseudo-Riemannian manifolds.
Categories: math.DG gr-qc math-ph math.MP