An essentially saturated surface not of Kahler type

It is shown that if X is an Inoue surface of type S_M then theirreducible components of the Douady space of Xⁿ are compact,for all n 0. This gives an example, sought by Moosa J. ReineAngew. Math. 586 (2005) 1–20], of an essentially saturatedcompact complex manifold (in the sense of model theory) thatis not of Kähler type. Among the known compact complexsurfaces without curves, these are the only examples.