Rigid Manifolds, Projective classifying spaces, Inoue type varieties and deformation to hypersurface embeddings

A classifying space is a space whose universal covering is contractible. There are not so many projective varieties which are PCS = projective classifying spaces, but many rigid surfaces of general type tend to be PCS. I shall discuss some joint work with Ingrid Bauer on rigid manifolds, and some surfaces for which it is still open the question whether they are rigid, respectively PCS. I shall then explain some examples how hyper surfaces in PCS lead to varieties whose moduli spaces can be understood through topology, as the Inoue type varieties, which are quotients of hyper surfaces in PCS. Strong rigidity of these depend on strong conditions, and one would like to enlarge the definition in the case of high degree. Studying the deformations of these, one is naturally lead to study finite maps $f$ which deform to hypersurface embeddings. After giving some examples, I shall explain some work in progress together with Yongnam Lee, characterising the case where the map $f$ has prime degree.