Classification of abelian Nash manifolds

TL;DR

This paper classifies abelian Nash manifolds by linking their structure to real abelian varieties, providing a comprehensive classification up to Nash equivalence.

Contribution

It establishes a classification of abelian Nash manifolds through their algebraization to real abelian varieties, extending the understanding of Nash manifolds.

Findings

Connected affine Nash groups are abelian Nash manifolds iff their algebraization is a real abelian variety.

Complete classification of real abelian varieties up to isomorphism.

Classification of abelian Nash manifolds up to Nash equivalence.

Abstract

By the algebraization of affine Nash groups, a connected affine Nash group is an abelian Nash manifold if and only if its algebraization is a real abelian variety. We first classify real abelian varieties up to isomorphisms. Then with a bit more efforts, we classify abelian Nash manifolds up to Nash equivalences.