what are some examples of Abelian varieties?

Question

I am looking at Milne's notes on Abelian varieties. Elliptic curves have an equation:

$$ y^2 z = x^3 + a x z + b z^3 \text{ and }4a^3+27b^2 \neq 0$$

but also as complex manifold. $E(\mathbb{C}) = \mathbb{C}/\Lambda$. For Abelian varieties, there is no equation for dimension > 1. That's fine.

Product two elliptic curves is an Abelian variety. OK.

What was original motivation for Abelian varieties? What are some situations that motivate study of Abelian varieties?

Answer 1

As requested by Alex Youcis I have created an answer based on the comments. To be expanded into an answer as I understand it.

@Gunnar Þór Magnússon

I believe the original motivation comes from the Jacobian varieties of compact Riemann surfaces, which are abelian varieties. An abelian variety is projective, but there are lots of non projective tori. –

You can write down a non-algebraic torus pretty easily ;) https://math.stackexchange.com/a/36658/3225

@ted

Not sure what your comments are asking, but maybe the point is that the isomorphism class of an abelian variety is not determined by just the underlying topological space. There is also the holomorphic/algebraic structure. This is true even for elliptic curves. All elliptic curves over $\mathbb{C}$ are homeomorphic to a torus, topologically, but they are not all isomorphic as elliptic curves. –

@Alexyoucis

I second Gunnar's comment. While there are many good reasons to consider abelian varieties, I think that thinking about trying to have a notion of 'Picard geometry' (to geometrize the Picard group) is probably the most convincing naive motivation. Another perspective that, depending on your bend, could be motivating is thinking about Galois representations. Namely, one gets a slew of Galois representations by thinking about $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acting on the torsion of group varieties over $\mathbb{Q}$. One of the most natural sources for such objects are AVs.

It turns out that for complex tori, being an abelian variety is the same thing as being 'Moishezon'. Namely, if $X$ is a complex torus which is algebraic (meaning that it's locally cut out by polynomial equations) then a basic fact one can show is that the meromorphic function field $M(X)$ of $X$ has transcendence degree $\mathrm{dim}X$ over $\mathbb{C}$--this is a property called Moishezon. One can write down in dimensions $g>1$ tori which are not Moishezon. The connection to algebraic geometry is indicated by @GunnarÞórMagnússon Namely, all algebraic tori are projective, and so one can embed them

into projective space. Embeddings into projective space come from line bundles, and line bundles are certain types of 'submodules' (subsheaves) of $M(X)$. So, the easiest way to write down non-algebraic tori is to write down ones that don't have enough meromorphic functions to embed them into projective space. This is written quite nicely in the relevant chapter of Cornell-Silverman.