Articulated Zilber-Pink conjecture for mixed Shimura and semi-abelian varieties

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Let XX be a mixed Shimura variety or a semi-abelian variety defined over C\mathbb{C}. Let ⟨A⟩\langle A\rangle denote the smallest special subvariety containing a subvariety AA, and define its defect by

δ(A)=dim⁡⟨A⟩−dim⁡A.\delta(A)=\dim\langle A\rangle-\dim A.

For T∈SXT\in\mathcal{S}_X, V⊆TV\subseteq T, and a non-negative integer δ\delta, let Atyp δ(V,T){\rm Atyp\,}^{\delta}(V,T) be the union of subvarieties A⊆VA\subseteq V with δ(A)≤δ\delta(A)\leq\delta. Articulated Zilber-Pink conjecture. The set Atyp δ(V,T){\rm Atyp\,}^{\delta}(V,T) is a finite union of atypical subvarieties of defect at most δ\delta. The source explains that this is a stronger, defect-by-defect formulation implied formally by Zilber-Pink; the supplied text does not state that either conjecture is resolved.

References

Primary source

Philipp Habegger and Jonathan Pila, “O-minimality and certain atypical intersections”, arXiv:1409.0771 (2014).

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