Submission date: 22 September 2022

Abstract:

Let X ⊂ ℂ^n be an algebraic variety, and let Λ ⊂ ℂ^n be a discrete subgroup whose real and complex spans agree. We describe the topological closure of the image of X in ℂ^n/Λ, thereby extending a result of Peterzil-Starchenko in the case when Λ is cocompact.
We also obtain a similar extension when X ⊂ ℝ^n 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 ⊂ ℂ^n (such as a complex algebraic variety) and exp : ℂ^n → (ℂ^*)^n the coordinate-wise exponential map, we have exp(X)=exp(X) ∪ ⋃{i=1}^m exp(C_i)⋅ 𝕋_i where 𝕋_i ⊂ (ℂ^*)^n are positive-dimensional compact real tori and C_i ⊂ ℂ^n are semi-algebraic.

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Full text arXiv 2209.10812: pdf, ps.