The Coleman–Oort conjecture for special subvarieties in the Torelli locus

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Let Ag\mathcal{A}_g be the moduli space of principally polarized complex abelian varieties of dimension gg, let TgT_g be the Torelli locus, and let TgoT_g^o be its open Torelli locus. A subvariety S⊂AgS\subset\mathcal{A}_g is special if it is an irreducible component of a Shimura variety. Coleman–Oort conjecture. For sufficiently large gg, there are no positive-dimensional special subvarieties S⊂AgS\subset\mathcal{A}_g contained in TgT_g and meeting TgoT_g^o. This follows formally from Coleman's conjecture together with the André–Oort conjecture, but the stated consequence is not resolved in the supplied text.

References

Primary source

Thomas Bouchet, Jeroen Hanselman, Andreas Pieper and Sam Schiavone, “Mumford-type Shimura curves contained in the Torelli locus”, arXiv:2510.00093 (2025).

Additional references

7 papers in this index state this conjecture (2016–2025). The statement above is taken from the most recent of them; the others are arXiv:2201.11971, arXiv:2109.07261, arXiv:2102.12349, arXiv:1812.04341, arXiv:1809.06315, arXiv:1611.08477.

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