Consider a riemannian manifold M. The moduli space of riemannian metrics on M, that admit parallel spinors on its universal covering, modulo the action of the diffeomorphism group, is a smooth manifold of finite dimension. This additional structure raises intriguing questions about its geometry. In this thesis, we present an argument showing that, for the specific case of M being a K3 surface, this moduli space (which consists of Ricci-flat metrics) is not complete.
Guadalupe Castillo Solano (Thu,) studied this question.