Deformation theory of nearly K\"ahler manifolds

Nearly K"ahler manifolds are the Riemannian 6-manifolds admitting real Killing spinors. Equivalently, the Riemannian cone over a nearly K"ahler manifold has holonomy contained in G2. In this paper we study the deformation theory of nearly K"ahler manifolds, showing that it is obstructed in general. More precisely, we show that the infinitesimal deformations of the homogeneous nearly K"ahler structure on the flag manifold are all obstructed to second order.