o-minimal flows on abelian varieties

TL;DR

This paper characterizes the Zariski closure of images of unbounded o-minimal definable sets under the uniformization map of complex abelian varieties, linking model theory and algebraic geometry.

Contribution

It provides a new description of the Zariski closure of o-minimal definable sets in the context of complex abelian varieties, bridging model theory and algebraic geometry.

Findings

Describes the Zariski closure of $ ext{pi}(X)$ for o-minimal definable $X$

Connects o-minimal structures with algebraic properties of abelian varieties

Advances understanding of the interaction between definability and algebraic geometry

Abstract

Let $A$ be an abelian variety over ${\bf C}$ of dimension $n$ and $\pi\colon {\bf C}^n \rightarrow A$ be the complex uniformisation. Let $X$ be an unbounded subset of ${\bf C}^n$ definable in a suitable o-minimal structure. We give a description of the Zariski closure of $\pi(X)$.