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 $\dim A \geq 2$.)

  • 4
    $\begingroup$ In general h^1,0 >= dim A, but that doesn't use the fact that the map is a closed immersion. $\endgroup$
    – PRL
    Dec 2, 2012 at 18:11
  • 1
    $\begingroup$ Olivier Debarre has several papers on subvarieties of (simple) abelian varieties which you might find useful/interesting. $\endgroup$
    – naf
    Dec 3, 2012 at 4:19
  • 2
    $\begingroup$ In this case, $X$ in particular has maximal Albanese dimension. Green-Lazarsfeld ("Deformation theory...", Invent. Math. 90) showed that in this case $(-1)^{{\rm dim}(X)}\chi(X,{\cal O}_X)\geq 0$. $\endgroup$ Dec 3, 2012 at 7:33

1 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.

  • 3
    $\begingroup$ 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. $\endgroup$ Dec 3, 2012 at 7:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.