Question

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 $\dim A \geq 2$.)

Answer 1

At least over the complex numbers, $X$ is of general type by an old result of Ueno (see Damian's comment below) that says the following:

Let $E$ be the biggest abelian subvariety of $A$ such that $X$ is invariant under translation by $E$, then $X/E$ is of general type.

ADDED: (prompted by Damian's comment to the question). Pareschi and Popa [ Strong generic vanishing and a higher-dimensional Castelnuovo-de Franchis inequality. Duke Math. J. 150 (2009), no. 2, 269–285. ] have generalized the Castelnuovo-De Franchis inequality for surfaces and have proven that, if $X$ has no fibration onto a lower dimensional irregular variety, then $\chi(\omega_X)\ge q(X)-\dim X$, where $q(X)\ge \dim A$ is the irregularity. In particular, this applies if the Albanese variety of $X$ is simple.