I'm having trouble distinguishing the various sorts of tori.

One definition of torus is the algebraic torus. Groups like SU(2,ℂ/ℝ) and SU(3,ℂ/ℝ) have important subgroups that are topologically a circle and a torus, and I guess those were some of the most important Lie groups so the name torus stuck. Groups like SL(2,ℂ) and SL(n+1,ℂ) have a similar important subgroup isomorphic to ℂ* and (ℂ*)^{n}, so the name torus gets applied to them too. In general, one calls the multiplicative group of an arbitrary field a torus in many situations, sometimes denoting the entire lot of them as Gm.

Another definition of a topological torus is a direct product of circles. A standard way to construct various flat geometries on a torus is to take ℝ^{n} and quotient out by a discrete rank n lattice Λ, for instance ℝ/ℤ or ℂ/ℤ[i]. A complex torus is defined analogously as ℂ^{n}/Λ where Λ is a rank 2n lattice (since ℂ^{n} has real rank 2n).

One reads in various places that every abelian variety is a complex torus, but not every complex torus is an abelian variety. The notation ℂ^{n}/Λ is usually nearby.

Is the multiplicative group of the field, Gm or ℂ*, an abelian variety?

In other words, is an algebraic torus over the complexes a complex torus?

Is an abelian variety isomorphic as a group to ℂ

^{n}/Λ, or just topologically?

My dim memory of elliptic curves was that they were finitely generated abelian groups, but since they are uncountable that doesn't make any sense. Presumably I am thinking of their rational points. However, ℂ^{n}/Λ is always an abelian group, so I don't see what the fuss is about deciding when it is an abelian variety. It seems likely to me the group operations are different.

real toruswith a specific (flat)complex structure. $\mathbb{C}*^n$ is a complex(ified) torus. – some guy on the street Apr 6 '10 at 16:08Complex tori and Abelian varieties, by Olivier Debarre. There you'll find for example the Riemann conditions which are necessary for a torus to be abelian. – Mariano Suárez-Alvarez♦ Apr 6 '10 at 16:08