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Locally \(\mathbb {C}\)-Nash groups

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Abstract

We introduce the category of locally \({\mathbb {C}}\)-Nash groups, basic examples of such groups are complex algebraic groups. We prove that the latter form a full subcategory. We also show that both, abelian locally Nash and abelian locally \({\mathbb {C}}\)-Nash groups, can be characterised via meromorphic maps admitting an algebraic addition theorem; we give an invariant of such groups associated to the groups of periods of a chart at the identity. Finally, we prove that the category of simply connected abelian locally \({\mathbb {C}}\)-Nash groups coincides with that of universal coverings of the abelian complex irreducible algebraic groups (a complex version of a result of Hrushovski and Pillay in Isr J Math 85(1–3):203–262, 1994; Conflu Math 3(4):577–585, 2011).

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Authors and Affiliations

  1. Departamento de Álgebra, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040, Madrid, Spain

    Elías Baro

  2. Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Juan de Vicente & Margarita Otero

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  1. Elías Baro
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  2. Juan de Vicente
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  3. Margarita Otero
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Correspondence to Margarita Otero.

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Elías Baro and Margarita Otero are partially supported by Spanish MTM2014-55565-P and Grupos UCM 910444. Juan de Vicente also supported by a grant of the International Program of Excellence in Mathematics at Universidad Autónoma de Madrid.

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Baro, E., de Vicente, J. & Otero, M. Locally \(\mathbb {C}\)-Nash groups. Rev Mat Complut 32, 531–558 (2019). https://doi.org/10.1007/s13163-018-0278-1

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