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


One definition of torus is the algebraic torus. Groups like $SU(2,\mathbb{C})$ and $SU(3,\mathbb{C})$ 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 $SL(2,\mathbb{C})$ and $SL(n+1,\mathbb{C})$ have a similar important subgroup isomorphic to $\mathbb{C}^\ast$ and $(\mathbb{C}^\ast)^n$, 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 $\mathbb{G}\_m$.


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 $\mathbb{R}^n$ and quotient out by a discrete rank $n$ lattice $\Lambda$, for instance $\mathbb{R}/\mathbb{Z}$ or $\mathbb{C}/\mathbb{Z}[i]$. A complex torus is defined analogously as $\mathbb{C}^n/\Lambda$ where $\Lambda$ is a rank $2n$ lattice (since $\mathbb{C}^n$ 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 $\mathbb{C}^n/\Lambda$ is usually nearby.



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> Is the multiplicative group of the field, $\mathbb{G}\_m$ or $\mathbb{C}^\ast$, an abelian variety?
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In other words, is an algebraic torus over the complexes a complex torus?



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> Is an abelian variety isomorphic as a group to $\mathbb{C}^n/\Lambda$, or just topologically?
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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, $\mathbb{C}^n/\Lambda$ 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.



Complex torus, C^n/Λ versus (C*)^n
The difference between an Abelian variety and $\mathbb{C}^n/\Lambda$ is that an abelian variety is *polarized*; that is, it comes with an ample line bundle, which yields an embedding into $\mathbb{P}^m$ for some $m$.


That is, Abelian varieties are projective algebraic, whereas complex tori (in the sense of $\mathbb{C}^n/\Lambda$) are not necessarily.


The fact that we also call $\mathbb{C}^\*$ a torus is, to the best of my knowledge, unrelated. It is not an Abelian variety.


