# subvarieties of abelian varieties over number fields

How "special" are closed subvarieties of abelian varieties over number fields? (Dimension 1 is easy.)

For example: Are there interesting families of varieties of general type which are not closed subvarieties of abelian varieties?

This is motivated by Faltings' articles Diophantine approximation on abelian varieties. Ann. of Math. (2) 133 (1991), no. 3, 549–576 and The general case of S. Lang's conjecture. Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991), 175–182, Perspect. Math., 15, Academic Press, San Diego, CA, 1994.: To which varieties does Faltings' theorem apply?

(There is the related question Properties of subvarieties of a simple abelian variety.)

• What do you mean by "interesting"? Simply connected varieties, for instance hypersurfaces, complete intersections... do not map non-trivially to any abelian variety. – abx Jan 23 '18 at 16:09

A smooth hypersurface of degree $d\geq 1$ in $\mathbb{P}^{n+1}_{\mathbb{C}}$ is simply connected when $n\geq 2$, as abx comments.
• Thanks. Hypersurfaces $X \hookrightarrow \mathbf{P}^n_\mathbf{C}$ are simply connected for $n \geq 3$ because this induces an injection on fundamental groups. – TKe Jan 23 '18 at 17:37