Geometry of a desingularization of eleven-dimensional gravitational spinors

We show that the space of gravitational spinors in eleven dimensions, defined by equations ${\mathrm{\Gamma }}_{\alpha \beta }^i{\lambda }^{\alpha }{\lambda }^{\beta }=0$ admits a desingularization with nice geometric properties. In particular the desingularization fibers over the isotropic Grassmannian OGr(2,11). This enables us to recast equations of linearized eleven-dimensional supergravity adapted to 3-form potential into Cauchy-Riemann equations on a super extension of isotropic Grassmannian OGr(2,11).