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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.)

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    $\begingroup$ What do you mean by "interesting"? Simply connected varieties, for instance hypersurfaces, complete intersections... do not map non-trivially to any abelian variety. $\endgroup$
    – abx
    Commented Jan 23, 2018 at 16:09

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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.

More generally, a simply connected positive-dimensional smooth projective variety is not a subvariety of an abelian variety. Thus, K3 surfaces are not subvarieties of abelian varieties, and positive-dimensional Fano varieties are not subvarieties of abelian varieties. With the appropriate definition of "Calabi-Yau", positive-dimensional subvarieties of abelian varieties are not Calabi-Yau.

Basically, subvarieties of abelian varieties are the varieties with a "large" Albanese variety.

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  • $\begingroup$ Thanks. Hypersurfaces $X \hookrightarrow \mathbf{P}^n_\mathbf{C}$ are simply connected for $n \geq 3$ because this induces an injection on fundamental groups. $\endgroup$
    – user19475
    Commented Jan 23, 2018 at 17:37

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