Albanese maps for open algebraic spaces

TL;DR

This paper extends the concept of Albanese maps to open algebraic spaces, establishing the existence of universal morphisms to para-abelian varieties and analyzing their properties, especially over imperfect fields.

Contribution

It generalizes classical Albanese map results to a broader class of algebraic spaces, including non-proper and imperfect field cases.

Findings

Existence of universal morphisms to para-abelian varieties for separated algebraic spaces.

The formation of Albanese varieties commutes with base change up to universal homeomorphisms.

Detailed analysis of Albanese maps for algebraic curves and groups over imperfect fields.

Abstract

We show that for each algebraic space that is separated and of finite type over a field, and whose affinization is connected and reduced, there is a universal morphism to a para-abelian variety. The latter are schemes that acquire the structure of an abelian variety after some ground field extension. This generalizes classical results of Serre on universal morphisms from algebraic varieties to abelian varieties. Our proof relies on corresponding facts for the proper case, together with the structural properties of group schemes, removal of singularities by alterations, and ind-objects. It turns out that the formation of the Albanese variety commutes with base-change up to universal homeomorphisms. We also give a detailed analysis of Albanese maps for algebraic curves and algebraic groups, with special emphasis on imperfect ground fields.