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?



"Simple" Kahler manifolds
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>


