(Joint work with Klaus Kroencke, Hartmut Weiss and Frederik Witt.) We study the set of all Ricci-flat Riemannian metrics on a given compact manifold $M$. We say that a Ricci-flat metric on $M$ is structured if its pullback to the universal cover admits a parallel spinor. The holonomy of these metrics is special as these manifolds carry some additional structure, e.g. a Calabi-Yau structure or a $G_2$-structure. 

 

The set of unstructured Ricci-flat metrics is poorly understood. Nobody knows whether unstructured compact Ricci-flat Riemannian manifolds exist, and if they exist, there is no reason to expect that the set of such metrics on a fixed compact manifold should have the structure of a smooth manifold. 

 

On the other hand, the set of structured Ricci-flat metrics on compact manifolds is now well-understood. 

 

The set of structured Ricci-flat metrics is an open and closed subset in the space of all Ricci-flat metrics. The holonomy group is constant along connected components. The dimension of the space of parallel spinors as well. The structured Ricci-flat metrics form a smooth Banach submanifold in the space of all metrics. Furthermore the associated premoduli space is a finite-dimensional smooth manifold. 

 

Our work builds on previous work by J. Nordstr\"om, Goto, Tian \& Todorov, Joyce, McKenzie Wang and many others. The important step is to pass from irreducible to reducible holonomy groups.