Group law, cubics and Lie group

Question

Let $C$ be a smooth complex cubic in $CP^{2}$. We know that there is a group structure by using the intersection of projective lines (cf. Ried, Undergraduate AG, Section 2), which is really different from the usual Lie group structure of 2-torus. My question is whether this group structure makes a Lie group? Or even a topological group under the usual complex topology? Thanks in advance.

Answer 1

Yes, it can be proven that any elliptic curve over $\mathbb C$ is isomorphic to $\mathbb C / \Lambda $ for some lattice $\Lambda \subset \mathbb C$ (a lattice is a additive subgroup of $\mathbb C$ generated by two elements $\omega_1, \omega_2)$.

But since any lattice is, after choosing a basis, isomorphic to $\mathbb Z^2$, and $\mathbb C = \mathbb R^2$, it follows that $\mathbb C / \Lambda \simeq \mathbb R/\mathbb Z \times \mathbb R /\mathbb Z \simeq S^1 \times S^1$, which is topologically a torus.

So an elliptic curve is in fact isomorphic to a torus as a Lie group. But there are many complex structures on $\mathbb C/\Lambda$ (corresponding to different $\omega_1,\omega_2)$, so they are not all isomorphic as complex manifolds.