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?

simplein many branches of mathematics. An object issimpleif it does not contain non-trivially another object of the same type. Simple groups are far from being simple. No one claims that that terminology is "contradictory". – Sándor Kovács Feb 23 '11 at 22:55