  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).
