Let $A$ be a simple abelian variety over a field $k$. (For simplicity, we assume char $k =0$.)
Let $X$ be a smooth projective geometrically connected variety over $k$ of positive dimension.
Suppose that there exists a closed immersion $X\to A$. What can we say about $X$?
If $\dim X=1$, it follows that the genus of $X$ is at least $\dim A$. (In fact, if $g(X) >0$, this follows from the universal property of the Albanese variety and Poincaré's irreducibility theorem. If $g(X) = 0$, there are no non-constant maps $X\to A$.)
What if $\dim X =2$? Can we say something about the "genus" of $X$? (Of course, here I assume $\dim A \geq 2$.)