Search

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. [ABSTRACT FROM AUTHOR]