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

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.