I previously asked this question on [MSE](https://math.stackexchange.com/q/1748216/39599) and offered a bounty but received no responses.




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There are examples of compact complex manifolds with no positive-dimensional compact complex submanifolds. For example, generic tori of dimension greater than one have no compact complex submanifolds. The proof of this fact, see [this answer](https://math.stackexchange.com/a/36658/39599) for example, shows that these tori also have no positive-dimensional analytic subvarieties either (because analytic subvarieties also have a fundamental class). 


My question is whether the non-existence of compact submanifolds always implies the non-existence of subvarieties.



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> Does there exist a compact complex manifold which has positive-dimensional analytic subvarieties, but no positive-dimensional compact complex submanifolds?
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Note, any such example is necessarily non-projective.



Complex manifold with subvarieties but no submanifolds
There are surfaces of type $VII\_0$ on which the only subvariety is a nodal rational curve (I. Nakamura, Invent. math. 78, 393-443 (1984), Theorem 1.7, with $n=0$).


