Locally C-Nash groups

TL;DR

This paper introduces the concept of locally C-Nash groups, explores their properties, and establishes connections with complex algebraic groups, meromorphic maps, and universal coverings, providing new insights into their structure and classification.

Contribution

It defines locally C-Nash groups, proves their isomorphism properties with algebraic groups, and characterizes abelian cases via meromorphic maps with algebraic addition theorems.

Findings

Locally C-Nash groups are closely related to complex algebraic groups.

Isomorphic locally C-Nash groups are also biregularly isomorphic.

The category of simply connected abelian locally C-Nash groups matches universal coverings of algebraic groups.

Abstract

We introduce the category of locally C-Nash groups, basic examples of such groups are complex algebraic groups. We prove that if two complex algebraic groups are locally C-Nash isomorphic then they also are biregularly isomorphic. We also show that both, abelian locally Nash and abelian locally C-Nash groups, can be characterized 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 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).