Mapping class groups of complex complete intersections, I

The mapping class group of closed complex curves is a central object in modern geometry and was intensively studied and plays a central role in the study of moduli spaces. But also today there are open questions. I want to review this briefly. This is the background for recent work with Su Yang about the mapping class group of certain complex 3-dimensional varieties, namely complete intersections. There is a surprising analogy between the case of complex curves and these manifolds in so far as in both cases there are analogous short exact sequences determining the mapping class groups. But there is a difference, the last exact sequence contains in the case of complex curves a mysterious unknown group, whereas we can compute the corresponding group for 3-dimensional complete intersections completely. The techniques for the proofs are partly similar but in the end we apply tools which are only available in higher dimensions, which I will explain. These methods are in principal useful in general so one learns something of general interest. In the end I will briefly report about applications of our results by Oscar Rendall-Williams to the moduli space of complex hypersurfaces.T