Pure spinors, intrinsic torsion and curvature in even dimensions

We study the geometric properties of a 2m  -dimensional complex manifold MM admitting a holomorphic reduction of the frame bundle to the structure group P⊂Spin(2m,C)P⊂Spin(2m,C), the stabiliser of the line spanned by a pure spinor at a point. Geometrically, MM is endowed with a holomorphic metric g, a holomorphic volume form, a spin structure compatible with g, and a holomorphic pure spinor field ξ up to scale. The defining property of ξ is that it determines an almost null structure, i.e. an m  -plane distribution NξNξ along which g is totally degenerate.We develop a spinor calculus, by means of which we encode the geometric properties of NξNξ corresponding to the algebraic properties of the intrinsic torsion of the P-structure. This is the failure of the Levi-Civita connection ∇ of g to be compatible with the P-structure. In a similar way, we examine the algebraic properties of the curvature of ∇.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 (M,g)(M,g) has prescribed curvature.We discuss applications of this work to the study of real pseudo-Riemannian manifolds.