Jacobian of hyper elliptic curve and $\Bbb{C}^g/Λ$

Question

Let $C$ be an genus $g$ hyper elliptic curve.

Let $J(C)$ be an Jacobi variety of $C$. Then, it is know that there is isomorphism $A \cong \Bbb{C}^g/Λ$, where $Λ$ is lattice.

I have two questions about this.

$1.$ $\cong $ ís as what ? This is certainly isomorphism as Riemann surface, but also group isomorphism ? (In my ambiguous memory, someone said that abelian variety except for elliptic curve does not have uniformaization in general.So I feel weird if Jacobin of hyperelliptic curve can have uniformization $A \cong \Bbb{C}^g/Λ$.)

$2.$What looks like $\Bbb{C}^g/Λ$ as topological space ? I can only imagine $d=1$ case, that is donut.Can we descrive　$\Bbb{C}^g/Λ$　in our head as topological space ?

Answer 1

For any abelian variety (and actually complex compact lie group), the isomorphism $A \cong \mathbb{C}^g/\Lambda$ is an isomorphism of complex lie groups. In fact, more is true:

A holomorphic map between complex tori is a composition of a holomorphic homomorphism and a translation.

As for your second question, as real lie groups, $$\mathbb{C}^{g}/\Lambda \cong \mathbb{R}^{2g}/\mathbb{Z}^{2g} \cong (S^1)^{2g} = \mathbb{T}^g$$

where $\mathbb{T}$ is the usual torus. You can look at Griffiths and Harris 'Principles of Algebraic Geometry' page $325$ for some details.