Algebraic and o-minimal flows beyond the cocompact case

Let $X\subset {\mathbb{C}}^n$ be an algebraic variety, and let $\mathrm{\Lambda }\subset {\mathbb{C}}^n$ be a discrete subgroup whose real and complex spans agree. We describe the topological closure of the image of $X$ in ${\mathbb{C}}^n\mathrm{/}\mathrm{\Lambda }$, thereby extending a result of Peterzil-Starchenko in the case when $\mathrm{\Lambda }$ is cocompact. We also obtain a similar extension when $X\subset {\mathbb{R}}^n$ is definable in an o-minimal structure with no restrictions on $\mathrm{\Lambda }$, and as an application prove the following conjecture of Gallinaro: for a closed semi-algebraic $X\subset {\mathbb{C}}^n$ (such as a complex algebraic variety) and $\exp :{\mathbb{C}}^n\to ({\mathbb{C}}^{\ast }{)}^n$ the coordinate-wise exponential map, we have $\overline{\exp (X)}=\exp (X)\cup {\bigcup }_{i=1}^m\exp (C_i)\cdot {\mathbb{T}}_i$ where ${\mathbb{T}}_i\subset ({\mathbb{C}}^{\ast }{)}^n$ are positive-dimensional compact real tori and $C_i\subset {\mathbb{C}}^n$ are semi-algebraic.