I previously asked this question on MSE and offered a bounty but received no responses.
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 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.
Does there exist a compact complex manifold which has positive-dimensional analytic subvarieties, but no positive-dimensional compact complex submanifolds?
Note, any such example is necessarily non-projective.
Best Answer
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 $n=0$).