2022/09/22 by Spencer Dembner, Hunter Spink, Dembner, Spencer +1
Mathematics · #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2209.10812
openalex publication_date 2022/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X ⊂ ℂn be an algebraic variety, and let Λ⊂ ℂn be a discrete subgroup whose real and complex spans agree. We describe the topological closure of the image of X in ℂn / Λ, thereby extending a result of Peterzil-Starchenko in the case when Λ is cocompact. We also obtain a similar extension when X⊂ ℝn is definable in an o-minimal structure with no restrictions on Λ, and as an application prove the following conjecture of Gallinaro: for a closed semi-algebraic X⊂ ℂn (such as a complex algebraic variety) and exp:ℂn→ (ℂ^*)n the coordinate-wise exponential map, we have exp(X)=exp(X)∪ \bigcupi=1m exp(Ci)⋅ \mathbbTi where \mathbbTi⊂ (ℂ^*)n are positive-dimensional compact real tori and Ci⊂ ℂn are semi-algebraic.