"Jacobian variety" for surfaces

Question

I heard that there isn't a functorial construction which associates an abelian variety to any 2-dimensional variety, equipped with an embedding of the surface inside the abelian variety. I find that a bit weird because this isn't the case for curves, for curves we have the Jacobian variety and the Abel-Jacobi map, which gives an embedding of the curve inside its Jacobian.

I wanted to know why this is the case? I.e., what is the obstruction for such a construction in the 2-dimensional case? What part in the construction of the Jacobian fails for higher dimensional varieties, such as surfaces?

I do know that some surfaces cannot be embedded inside any abelian manifold. Is it also true that there exist surfaces which cannot be embedded inside any group variety?

Thanks in advance!

Answer 1

Let $X$ be a normal projective variety.

1. If $X$ embeds into a group variety, then it maps finitely to an abelian variety. (Probably $X$ even embeds into an abelian variety.)

Indeed, suppose that $G$ is a group variety containing $X$. Replacing $G$ by its connected component, we may assume that $G$ is connected. Then, by Chevalley's theorem, we have a short exact sequence $0\to H\to G\to A\to 0$ with $H$ linear and $A$ an abelian variety. The intersection of $H$ and $X$ is an affine variety (because it is closed in $H$) which is also projective (because it is closed in $X$). Thus, it is a finite set.

This implies that the morphism $X\subset G\to A$ is finite, as required.

2. A simply connected smooth projective variety $X$ of dimension $>0$ does not map finitely to any abelian variety. (Consider fundamental groups or $\mathrm{H}^1(X,\mathcal{O}_X)$ to see this.)