Complex torus, C^n/Λ versus (C*)^n

Question

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.

Answer 1

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.

Answer 2

Let $ M=\mathbb C^n/Δ$ be a complex torus, then $M$ is abelian if and only if there is an integral closed positive $(1,1)$-current $\omega$ on $M$ and a point $p\in M $ such that $\omega−\epsilon\Sigma ^n_{i=1}dz_i∧d\bar z_i\geq0$ on $U$ in the sense of currents for some neighborhood $U$ of $p$ and for some positive constant number $\epsilon$, where $(z_1,⋯,z_n)$ is a coordinate system on $U$.

Note that the definition of positivity of current(introduced by Lelong) is different with positivity of a form

See Smoothing of currents and Moišezon manifolds. Several complex variables and complex geometry, by Ji, Shanyu

Answer 3

There are a number of things floating around here.

First among them is the first excellent point that Marino made that the finite generation of group of rational points of an abelian variety over a field K is only true for global fields. So let's say we're working over $\mathbf{C}$, where any positive dimensional variety has uncountably many points.

Second is the other excellent point of Marino that $\mathbf{C}^\times$ is not compact, so it can't fit with the definition of an abelian variety as a complete, connected group variety.

Third, it's much stronger to say that an abelian variety over the complex numbers is $\mathbf{C}^n/\Lambda$ topologically than group-theoretically. But in fact much more is true. Analytically, an abelian variety is isomorphic to $\mathbf{C}^n/\Lambda$. This comes from showing that the exponential map from the tangent space at the identity is in fact surjective, followed by figuring out the kernel. Details on this can be found in Milne's notes on abelian varieties or the first chapter of Mumford's book. In fact, even if we relax down to $C^\infty$ isomorphisms (let alone homeomorphisms) we could say that an abelian variety is isomorphic to $\mathbf{C}^n/\mathbf{Z}^{2n}$.

Answer 4

The reason why ${\mathbb C}^*$ is called torus is clear by looking at it real points. It could be a split real torus ${\mathbb R}_+^*$ or a compact torus $U_1({\mathbb C}) \cong {\mathbb R}/{\mathbb Z}$.

In general, a torus in a real Lie group looks like $({\mathbb R}_+^*)^n \times {\mathbb R}^m/{\mathbb Z}^m$. The split part $({\mathbb R}_+^*)^n$ is a bit like $({\mathbb C}^*)^n$. The compact part ${\mathbb R}^m/{\mathbb Z}^m$ is a topological torus.