Non projective hyperbolic compact complex space

Question

A famous conjecture by Kobayashi (perhaps slightly revisited subsequently) states that every compact hyperbolic Kähler manifold $X$ has ample canonical bundle.

This implies in particular that $X$ is projective (and canonically polarized...).

Here is my innocent-sounding question: is there any example of hyperbolic compact complex manifold which is not projective nor Kähler? What about if one allows singularities?

In other words, is the Kähler assumption necessary in the above-mentioned conjecture?

Thanks a lot in advance!

Answer 1

The non-projective Kahler surfaces are either K3, tori or elliptic surfaces, known to be non-hyperbolic. Non-Kahler surfaces are either elliptic or class VII; the elliptic surfaces are obviously non-hyperbolic. Minimal class VII surfaces with $b_2=0$ are either Hopf (obviously non-hyperbolic) or Inoue, by Bogomolov's theorem; the latter admit a holomorphic foliation with $\Bbb C$-fibers, hence non-hyperbolic. Minimal class VII surfaces with $b_2>0$ are conjecturally all Kato surfaces. The Kato surfaces (by Dlousky's theorem, I think) are obtained as limits of blown-up Hops surfaces, hence non-hyperbolic. For non-Kato surfaces, nothing is known, but they don't exist at least for $b_2=1$ by Teleman's result. I would expect that they are also non-hyperbolic, even if they exist (though everybody I know believes that non-Kato class VII${}_0$ surfaces don't exist, me included; much evidence supporting this view was collected by Teleman, Dlousky, Oeljeklaus, Toma and others).