TITLE: Complex analytic space with no (positive dim.) subscheme ?
QUESTION [7 upvotes]: Is there an example of a complex analytic space $X$ that doesn't have any (not necessarily open or closed) positive dimensional subspace $Y$ which is analytically isomorphic to (the complex analytic space associated to) a scheme?

Edit: after D.Arapura's comment, we require $X$ to have dimension $>1$. 
If I remember correctly, there are non "Abelian" complex tori $X=\mathbb{C}^n/\mathrm{Lattice}\;\;$ that do not have any positive dimensional analytic subvariety. Can a counterexample be derived from this?
Also, any complex algebraic space has an open subspace which is a scheme. So the counterexample (if it exists) must be searched outside algebraic spaces.

What if the question is modified by requiring that $X$ has no $Y$ that is locally closed in the analytic Zariski topology (where opens are complements of analytic subspaces)?

REPLY [2 votes]: Take a complex 2-torus $X$ without curves, and hence, without non-constant meromorphic functions (see e.g. Shafarevich, Basic algebraic geometry, chapter VIII, \S 1, example 2). The only locally closed subsets of $X$ will be $X$ itself, $\varnothing$, finite subsets and the complements of finite subsets. $X$ minus a finite subset can't be algebraic: if it were, it would have a non-constant meromorphic function, and then so would $X$.
[upd: locally closed here means locally closed in the analytic Zariski topology, so this answers the second question and not the first.]