On morphisms between connected commutative algebraic groups over a field of characteristic $0$

TL;DR

This paper investigates morphisms between connected commutative algebraic groups over a field of characteristic zero, establishing a natural retraction to group homomorphisms and characterizing automorphisms under certain conditions.

Contribution

It constructs a natural retraction from variety morphisms to group homomorphisms and provides explicit descriptions of morphisms and automorphisms for specific classes of algebraic groups.

Findings

A natural retraction from morphisms to homomorphisms is constructed.

If groups are isomorphic as varieties, they are isomorphic as groups.

Characterization of automorphisms as compositions of group automorphisms and translations.

Abstract

Let $K$ be a field of characteristic $0$ and let $G$ and $H$ be connected commutative algebraic groups over $K$. Let $\text{Mor}_0(G,H)$ denote the set of morphisms of algebraic varieties $G \to H$ that map the neutral element to the neutral element. We construct a natural retraction from $\text{Mor}_0(G,H)$ to $\text{Hom}(G,H)$ (for arbitrary $G$ and $H$) which commutes with the composition and addition of morphisms. In particular, if $G$ and $H$ are isomorphic as algebraic varieties, then they are isomorphic as algebraic groups. If $G$ has no non-trivial unipotent group as a direct factor, we give an explicit description of the sets of all morphisms and isomorphisms of algebraic varieties between $G$ and $H$. We also characterize all connected commutative algebraic groups over $K$ whose only variety automorphisms are compositions of automorphisms of algebraic groups with translations.