Lets parametrize the set of lattices inside $\mathbb C^g$ with the open dense subset $U= GL_{2g}(\mathbb R)$ of $\mathbb R^{4g^2}$.
Show that there exists a coutable family $(Z_n)_{n \in \mathbb N}$ of algebraic real hypersurfaces of $\mathbb R^{4g^2}$ such that for every matrix $M \in U$ \ $\bigcup_{n \in \mathbb N}Z_n $ the only complex subtori of $X = \mathbb C^g/\Gamma_M$ are $X$ and $\{0\}$.
I don't get how there cannot be for example a one dimensional subtorus of a $2$ dimensional complex torus. Given an invertible $4 \times 4$ real matrix, the first two columns are linearly independent, they generate a sublattice of real dimension $2$ stable under moltiplication by $i$. Why this is not a subtorus?
I know this "How to construct a simple complex torus of dimension $\geq 2$?" is related to the question, but the solution I'm interested is only given by a small hint in a comment.
Thanks!