Pink's Zilber–Pink conjecture for abelian varieties
Let be an algebraically closed field, let be an abelian variety defined over , and let be a subvariety of . A subvariety of is atypical for in if it is an irreducible component of the intersection of with a special subvariety of codimension at least ; it is maximal if it is not contained in any larger atypical subvariety.
Zilber's atypical-intersection conjecture. The subvariety 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).
Progress summary
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Solutions 1
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Claimed by OpenAI.
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
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