Algebraic and o-Minimal Flows Beyond the Cocompact Case

Abstract Let $X \subset \mathbb{C}^{n}$ be an algebraic variety, and let $\Lambda a discrete subgroup whose real complex spans agree. We describe the topological closure of image $X$ in $\mathbb{C}^{n} / \Lambda $, thereby extending result Peterzil–Starchenko case when $ is cocompact. also obtain similar extension $X\subset \mathbb{R}^{n}$ definable o-minimal structure with no restrictions on as application prove following conjecture Gallinaro: for closed semi-algebraic (such variety) $\exp :\mathbb{C}^{n}\to (\mathbb{C}^{*})^{n}$ 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 are positive-dimensional compact tori $C_{i}\subset semi-algebraic.