Algebraic and o-minimal flows beyond the cocompact case

TL;DR

This paper extends the understanding of the topological closure of algebraic and definable sets under certain quotient and exponential maps, generalizing previous results beyond cocompact cases and confirming a conjecture about exponential images.

Contribution

It generalizes the description of closures of algebraic and definable sets in complex and real spaces beyond cocompact lattices, and proves a conjecture on exponential images of semi-algebraic sets.

Findings

Extended closure descriptions to non-cocompact lattices in complex spaces.

Proved the Gallinaro conjecture on exponential images of semi-algebraic sets.

Unified algebraic and o-minimal approaches for closure characterizations.

Abstract

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.