Defect condition for mixed Shimura varieties and weakly special subvarieties

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Let XX be a mixed Shimura variety, and let A⊆B⊆XA\subseteq B\subseteq X be subvarieties. Write ⟨A⟩\langle A\rangle and ⟨B⟩\langle B\rangle for their smallest special subvarieties and ⟨A⟩geo\langle A\rangle_{\rm geo} and ⟨B⟩geo\langle B\rangle_{\rm geo} for their smallest weakly special subvarieties. Define

δ(A)=dim⁡⟨A⟩−dim⁡A,δgeo(A)=dim⁡⟨A⟩geo−dim⁡A.\delta(A)=\dim\langle A\rangle-\dim A, \qquad \delta_{\rm geo}(A)=\dim\langle A\rangle_{\rm geo}-\dim A.

Defect condition. Every mixed Shimura variety, and every weakly special subvariety, has the defect condition. The supplied span does not give the formal definition of the defect condition or state whether this claim is open, so the precise intended assertion requires checking against the surrounding source.

References

Primary source

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

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