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

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

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

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

Is the multiplicative group of the field, Gm$\mathbb{G}_m$ or ℂ*$\mathbb{C}^\ast$, 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 n$\mathbb{C}^n/\Lambda$, 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, n$\mathbb{C}^n/\Lambda$ 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.

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 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 n where Λ is a rank 2n lattice (since 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 n 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 n, 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, n 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.

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.

Is the multiplicative group of the field, $\mathbb{G}_m$ or $\mathbb{C}^\ast$, 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 $\mathbb{C}^n/\Lambda$, 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, $\mathbb{C}^n/\Lambda$ 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.

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Jack Schmidt
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Complex torus, C^n/Λ versus (C*)^n

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 ℝn 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 ℂn/Λ where Λ is a rank 2n lattice (since ℂn 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 ℂn/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 ℂn/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, ℂn/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.