The geometry of essentially saturated spaces

A compact complex space $X$ is called essentially saturated if and only if the components of the Douady space of each Cartesian power $X^n$ are compact; these spaces are of special interest in geometric model theory because they are universal domains for their first-order theories. Examples of essentially saturated spaces are given by spaces of Fujiki class $\mathcal{C}$ since the components of the Douady space of such spaces are always compact and the Cartesian product of two spaces of class $\mathcal{C}$ is again of class $\mathcal{C}$. The class $\mathcal{S}$ of essentially saturated spaces is thus a natural extension of the Fujiki class $\mathcal{C}$, and an interesting problem is to understand how these two classes differ. It is important to point out that not all compact complex spaces are essentially saturated; for example Hopf surfaces are not. Furthermore, one can show that there exist spaces of class $\mathcal{S}$ that are not of class $\mathcal{C}$. In this talk, I will give examples of essentially saturated spaces that are not of Fujiki class $\mathcal{C}$ and discuss some open questions.