Algebraic and o-Minimal Flows Beyond the Cocompact Case

Let $X \subset \mathbb{C}^{n}$ be an algebraic variety, and let $\Lambda \subset \mathbb{C}^{n}$ be a discrete subgroup whose real and complex spans agree. We describe the topological closure of the image of $X$ in $\mathbb{C}^{n} / \Lambda $, thereby extending a result of Peterzil–Starchenko in the case when $\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 $\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 $\exp :\mathbb{C}^{n}\to (\mathbb{C}^{*})^{n}$ 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 $\mathbb{T}_{i}\subset (\mathbb{C}^{*})^{n}$ are positive-dimensional compact real tori and $C_{i}\subset \mathbb{C}^{n}$ are semi-algebraic.