Let $G$ be a commutative connected algebraic group over $\mathbb{C}$. A theorem of Serre says that there exists an exact sequence
$$1\to \mathbb{G}\_a^n\times \mathbb{G}\_m^m\to G\to A\to 1,$$
where $A$ is an abelian variety. (See [here](https://icerm.brown.edu/materials/Slides/sp-s12-w3/On_Mordell-Lang_in_algebraic_groups_of_unipotent_rank_1_%5D_Paul_Vojta,_University_of_California,_Berkeley.pdf), for example.)


**I wonder what are some examples of such groups $G$ where this extension is non-trivial. That is, such that $G\ncong \mathbb{G}\_a^n\times \mathbb{G}\_m^m\times A$.**



What is a "non-trivial" example of a commutative algebraic group over $\mathbb{C}$?
One example is provided by the generalized Jacobian. For a smooth projective curve $C$ and a divisor $D$ on $C$, the generalized Jacobian is defined to be the moduli space parameterizing pairs consisting of a line bundle of degree $0$ on $C$ together with a trivialization of that line bundle over $D$.


This admits a map to the usual Jacobian, whose kernel is a product of $\mathbb G\_m$s and $\mathbb G\_a$s depending on the multiplicity of points of $D$.


If this were trivial, than we could define a section of this map, which would give for an arbitrary line bundle a canonical trivialization at each point of $D$. That would mean that for each point $x\in D$, the line bundle on the usual Jacobian whose fiber at a point $L \in J$ is the fiber of $L$ at $x$ would admit a section and thus be trivial. But from the duality theory of abelian varieties, these line bundles are different for distinct points $x\in C$, so they cannot all be trivial as long as $D$ contains two or more points.


