Given $\mathbb{C}^g/\Lambda$, when is it an algebraic variety? When $g=1$, there is a theorem that every one-dimensional Riemann surface is a projective algebraic variety, but what about higher dimensions?
I suspect this an overly hard question, but perhaps it will be eased into plausibility by with a particular $g$ and $\Lambda$ in mind.
Let $A$ be the three dimensional complex manifold $\mathbb{C}^3/\mathbb{Z}[\zeta_7]$ over $\mathbb{C}$ where $\mathbb{Z}[\zeta_7]$ is embedded in $\mathbb{C}^3$ by the following three homomorphisms: $$\sigma_1, \sigma_2, \sigma_3: \mathbb{Q}(\zeta_7) \to \mathbb{C} \times \mathbb{C} \times \mathbb{C}$$
Here, $\sigma_a$ is the homomorphism $\mathbb{Q}(\zeta_7) \to \mathbb{C}$ which sends $\zeta_7 \mapsto \zeta_7^\text{ a}$.
Q1. How can we see that $A$ is a complex algebraic variety?
I am not sure why $A$ admits a polarization, nor why polarizations are entirely relevant to answering Q1. From reading Hartshorne, I see that we have some options: we can prove that $A$ is Hodge, or prove that $A$ is both Kähler and Moishezon; both of these conditions imply projective algebraic variety. I haven't the slightest of how to show these things.
Q2. How do we derive the equations which cut out $A$?
Q3. In general, given $C^g/\Lambda$, when is it an algebraic variety?