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.

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

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 $\mathbf{C}^n/\mathbf{Z}^{2n}$.