Let us parametrize the set of lattices inside $\mathbb{C}^g$ with the open dense subset $U = \text{GL}\_{2g}(\mathbb{R})$ of $\mathbb{R}^{4g^2}$. Does there exist a countable family $(Z\_n)\_{n \in \mathbb{N}}$ of algebraic real hypersurfaces of $\mathbb{R}^{4g^2}$ such that for every matrix $M \in U \setminus \bigcup\_{n \in \mathbb{N}} Z\_n$, the only complex subtori of $X = \mathbb{C}^g/\Gamma\_M$ are $X$ and $\{0\}$?



A very general complex torus is simple
I think yes. Let $J$ be the complex structure on $\mathbb{R}^{2g}$. Let $L$ be the lattice. Then there is a complex subtorus if and only if $L$ intersects some complex-linear subspace in a full sub-lattice. Thus, there are no complex subtori if and only if for all $x\_1,\dots,x\_k\in L$ there do not exist $y\_1,\dots,y\_k\in L$ such that $$\textrm{span}\{x\_1,Jx\_1,x\_2,Jx\_2,\dots,x\_k,Jx\_k\}=\textrm{span}\{x\_1,y\_1,x\_2,y\_2,\dots,x\_k,y\_k\}.$$ This is countably many real-algebraic conditions. 


