An essentially saturated surface not of Kaehler-type

TL;DR

The paper demonstrates that certain Inoue surfaces of type SM have compact Douady spaces for all powers, providing examples of essentially saturated non-Kaehler compact complex manifolds, and identifies these as unique among curve-free surfaces.

Contribution

It introduces new examples of essentially saturated non-Kaehler surfaces by analyzing the Douady space of Inoue surfaces of type SM.

Findings

Douady space components of X^n are compact for all n>0

Inoue surfaces of type SM are essentially saturated and non-Kaehler

These are the only known curve-free compact complex surfaces with this property

Abstract

It is shown that if $X$ is an Inoue surface of type $S_M$ then the irreducible components of the Douady space of $X^n$ are compact, for all $n>0$. This gives an example of an essentially saturated compact complex manifold (in the sense of model theory) that is not of Kaehler-type. Among the known compact complex surfaces without curves, it is shown that these are the only examples.