"Simple" Kahler manifolds I have some lecture notes from Demailly on Kahler geometry where he talks about "variétés Kahleriennes simples", which are defined as Kahler manifolds $X$ such that for very generic points $x_0$ in $X$ there exists no irreducible submanifold $Y$ of $X$ of dimension $0 < \dim Y < \dim X$ which contains $x_0$.
Examples of these kinds of manifolds are very general complex tori and quotients thereof, and they're interesting because they give counterexamples to the Hodge conjecture in the analytic category.
I thought I'd take a look at these things, but I can't find any mention of "simple Kahler manifolds" either here or on google. Did I get the name wrong? Do any of you know what I'm talking about and know of some references?
 A: A generic deformation of a Hilbert scheme of K3 and a generic torus have no
subvarieties, hence they are "simple" in the above sense. For a torus it's
well known, for a Hilbert scheme of K3 it's in my paper
http://arxiv.org/abs/alg-geom/9705004
A: I've also never heard simple used in this way, so I expect you would have a hard time finding references. If you are really interested in pursuing this, you could ask Demailly. The condition looks interesting. Manifolds satisfying it are as far from projective algebraic manifolds as you can get. In particular, there are no nonconstant meromorphic functions on them. If you want another example, try a nonalgebraic K3 surface without an elliptic pencil.
One last comment: the Hodge conjecture is vacuously true for very general tori. The counterexamples, due to Voisin and Zucker, are tori but they are more subtle.
A: These manifolds have actually been studied to some extent by Campana and Peternell in their series of papers "Towards a Mori theory on compact Kähler threefolds I, II, III". In those papers they mention a folklore conjecture (which can be found for example here), that says that every simple Kähler manifold of odd dimension $n>1$ must be Kummer (i.e. bimeromorphic to the quotient of a complex torus by a finite group).
In the paper number II Peternell shows that this conjecture in dimension $3$ follows from MMP plus abundance for Kähler threefolds. On the other hand in paper number III he shows that abundance does hold for Kähler threefolds with the possible exception of simple non-Kummer manifolds (which should not exist).
On the other hand more recent developments using model theory seem to suggest that in the even-dimensional case apart from Kummer manifolds the only other simple Kähler manifolds are in generically finite-to-finite correspondence with an irreducible hyperkähler manifold, like in Misha's answer.
Anyway, even after all this work it's unclear whether simple non-Kummer odd-dimensional manifolds exist or not. It's certainly an interesting problem, but probably a very hard one.
A: This is a very interesting class of manifolds which, to my knowledge, has not been studied in any detail. One should be able to prove interesting structure theorems for such manifolds. For instance, I expect that the abelian category of analytic coherent sheaves on such manifolds is stable under deformation equivalence, i.e. if two such manifolds are deformation equivalent, they should have equivalent categories of coherent sheaves.
The one danger here is that the pool of examples may be very small. I guess, the first thing to look at is to find constructions of more examples. It will be interesting to find an example with a non-abelian infinte fundamental group. One can try to take a quotient of a torus by a freely acting finite group but I suspect that these are never simple.
Also, as Dmitry points out, the terminology is misleading, so if you are seriously thinking of working on these manifolds, now is the time to come up with a better name for them.
