Is there an intrinsic way to define the group law on Abelian varieties?

Question

On an elliptic curve given by a degree three equation y^2 = x(x - 1)(x - λ), we can define the group law in the following way (cf. Hartshorne):

1. We note that the map to its Jacobian given by $\mathcal{O}(p - p_0)$ for a fixed point $p_0$ is an isomorphism; ergo it inherits a group structure from the Jacobian.

2. In fact, if we embed it into $\mathbb{P}^2$ via the linear system $|3p_0|$, then three colinear points have $p + q + r \sim 3p_0$ and so this is in fact the group law inherited from Pic₀.

Is there an analogous way to do this for Abelian varieties? In Lange & Birkenhake they simply define an Abelian variety to be $\mathbb{C}^n$ modulo a lattice, and so it automatically comes with a group structure. Still, this seems unsatisfying in comparison to the way we can do so for elliptic curves.

That being said, the previous method doesn't seem to work for Abelian varieties; divisors no longer correspond to formal sums of points and so any comparison with Pic₀ wouldn't obviously yield a group structure on the points of X.

To make matters worse, the map that I tend to think of which takes X to Pic₀(X) is given by $p \mapsto t_p^*L \otimes L^{-1}$ for a given line bundle $L$ on X, where $t_p : X \to X$ is the map... defined by translation in X. So this map already requires the group structure on X to be defined.

So is there a way of defining the group law analogous to that of an elliptic curve?

NB: I do note that an elliptic curve can be defined as the zero locus of a cubic equation in $\mathbb{P}^2$, where I'm not sure how else we might define an Abelian variety other than as $\mathbb{C}^n$ modulo a lattice, and so perhaps the question is moot.

Answer 1

Any torsor $V$ under an abelian variety over any field $K$ is caonically isomorphic to its degree $1$ Albanese variety $\operatorname{Alb}^1(V)$, which is itself a torsor under the degree $0$ Albanese variety $\operatorname{Alb}^0(V)$. Note that the $\operatorname{Alb}^0$ of any smooth projective variety is a complete, geometrically connected group variety.

Assuming there exists a $K$-rational point $O$, one can subtract $O$ to obtain an isomorphism $\operatorname{Alb}^1(V) \stackrel{\sim}{\rightarrow} \operatorname{Alb}^0(V)$. Pulling back via the composite of these isomorphisms puts a group structure on $V$ depending only on the chosen base point $O$.

Answer 2

In dimension 1, the situation is like this: Every smooth proper genus 1 curve with a "marked" rational point has a unique group structure such that the given rational point becomes the neutral element.

For abelian varieties it is still true that the group structure, the origin being fixed, is unique: an abelian variety is canonically isomorphic to its Albanese variety, and to construct these you don't need the group structure.

Answer 3

For algebraically complete integrable systems Abelian varieties usually show up as follows: One has the preimage under n commuting integrals on a complex 2n-dim symplectic manifold. Then, for regular values, assuming that the integrals are proper, one has compact fibers whith n commuting holomorphic vectorfields (the symplectic gradients of the integrals) without zeros. This is an algebraic variety, and the group structure comes from its Lie algebra by flowing along the vectorfields.