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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 $\dim A \geq 2$.)

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In general h^1,0 >= dim A, but that doesn't use the fact that the map is a closed immersion. –  PRL Dec 2 '12 at 18:11
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Olivier Debarre has several papers on subvarieties of (simple) abelian varieties which you might find useful/interesting. –  ulrich Dec 3 '12 at 4:19
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In this case, $X$ in particular has maximal Albanese dimension. Green-Lazarsfeld ("Deformation theory...", Invent. Math. 90) showed that in this case $(-1)^{{\rm dim}(X)}\chi(X,{\cal O}_X)\geq 0$. –  Damian Rössler Dec 3 '12 at 7:33
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up vote 4 down vote accepted

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.

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The result of Ueno you are referring to is in K. Ueno, "Classification of algebraic varieties I", Compositio Math. 27, no. 3 (1973), Th. 3.10. See also Lang, "Survey of Diophantine geometry", I, par. 6, p. 35. –  Damian Rössler Dec 3 '12 at 7:50
    
@Damain: thank you! –  rita Dec 3 '12 at 9:15
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