  We study the geometric properties of a $2m$-dimensional complex manifold
$\mathcal{M}$ admitting a holomorphic reduction of the frame bundle to the
structure group $P \subset \mathrm{Spin}(2m,\mathbb{C})$, 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, ie 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 $\mathcal{N}_\xi$ 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. In particular, we
give necessary and sufficient conditions for the almost null structure
associated to a pure conformal Killing spinor to be integrable. We also
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.
