Inoue surfaces and Inoue type manifolds

Among minimal surfaces of general type with $p_g = 0$, there is only one class with $K^2 = 7$, discovered by Inoue 18 years ago. In recent work with Ingrid Bauer, we showed that these surfaces form an irreducible connected component of the moduli space, and that weak rigidity holds for them. Weak rigidity for $X$ means, in its weaker form, that every other variety $Y$ homotopically equivalent to $X$ has the property that either $Y$ or the conjugate variety belongs to an irreducible family containing $X$. We show this result by giving a different description of Inoue surfaces. This description lends itself to generalizations, which I will discuss during the talk. We define an Inoue type manifold as an ample divisor in a product of manifolds in the following list: Abelian varieties and quotients of irreducible locally symmetric spaces (including curves). Under some further assumptions, we can show semirigidity and weak rigidity results for Inoue type manifolds.