# Algebraic and o-minimal flows on complex and real tori

## TL;DR

This paper studies the topological closures of algebraic and definable sets under covering maps on complex and real tori, providing a unified description in both algebraic and o-minimal contexts.

## Contribution

It offers a new description of the topological closure of images of algebraic and definable sets under torus covering maps, bridging complex algebraic geometry and o-minimal structures.

## Key findings

- Describes closures of algebraic varieties in complex tori.
- Provides analogous results for definable sets in real tori.
- Unifies algebraic and o-minimal approaches to torus flows.

## Abstract

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.