The complex structure problem in geometry of curved spacetime

First, we’ll recall the complex structure problem in R. Penrose’s Twistor Programme. Then, we’ll principally analyse the spacetime geometry via contact structure.
In particular, we’ll consider possible construction for a global spinor bundle that should give the $\bar\partial$ operator on sections restricted to a sky. It uses so named reverse conformal reference frame, which is based on the contact $ST^*M$ bundle. These constructions will be connected with theory of gravitation.
Related to arXiv:1605.07070, to appear in Theor.Math.Phys.