Pink's Zilber–Pink conjecture for abelian varieties

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Let KK be an algebraically closed field, let AA be an abelian variety defined over KK, and let VV be a subvariety of AA. A subvariety WW of VV is atypical for VV in AA if it is an irreducible component of the intersection of VV with a special subvariety of codimension at least dim⁡V−dim⁡W+1\dim V-\dim W+1; it is maximal if it is not contained in any larger atypical subvariety.

Zilber's atypical-intersection conjecture. The subvariety VV contains at most finitely many maximal atypical subvarieties.

This is an unlikely-intersections formulation of the Zilber–Pink conjecture for abelian varieties, and it implies Pink's formulation; the paper uses it as a central conjectural principle. Its status in the stated generality is not specified by the source.

References

Primary source

Fabrizio Barroero and Gabriel Andreas Dill, “On the Zilber-Pink conjecture for complex abelian varieties”, arXiv:1909.01271 (2019).

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The manuscript claims finiteness of maximal atypical subvarieties for every irreducible subvariety of an abelian variety over a number field, with atypicality measured inside its smallest containing torsion coset. This records the number-field and special-closure formulation; it does not assert resolution of the target statement over arbitrary algebraically closed fields.

Repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Abelian-Zilber-Pink-Conjecture-September-24-2026/paper.pdf

  • OpenAI-016-01-The-abelian-Zilber-Pink-conjecture.pdf623,169 bytesOpen