Pure spinor formalism implies that supergravity equations in space-time are equivalent to the requirement that the worldsheet sigma-model satisfies certain properties. One of these properties has a particularly transparent geometrical interpretation: there exists an odd nilpotent vector field on some singular supermanifold, naturally associated to space-time. The normal form is parametrized by some tensor fields, which satisfy hyperbolic equations. These equations are similar to the full supergravity equations, but slightly weaker.