Algebraic and o-minimal flows on complex and real tori

We consider the covering map $\pi :{\mathbb{C}}^n\to \mathbb{T}$ of a compact complex torus. Given an algebraic variety $X\subseteq {\mathbb{C}}^n$ we describe the topological closure of $\pi (X)$ in $\mathbb{T}$. We obtain a similar description when $\mathbb{T}$ is a real torus and $X\subseteq {\mathbb{R}}^n$ is a set definable in an o-minimal structure over the reals.