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Conjecture on Intersections with Tori (CIT)

Prove the Conjecture on Intersections with Tori (CIT): For any algebraic variety W ⊆ K^n over Q in an algebraically closed field K of characteristic 0, construct a finite set μ(W) of proper algebraic subgroups of (K×)^n such that, for every proper algebraic subgroup A ⊆ (K×)^n, any atypical irreducible component S of W ∩ A (i.e., dim^K S > dim^K W + dim^K A − n) is contained in some subgroup B in μ(W).

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Background

The Conjecture on Intersections with Tori (CIT) is presented as a special case of the Zilber–Pink conjecture and predicts that atypical intersections with algebraic subtori are controlled by finitely many proper algebraic subgroups. In this paper, CIT underpins the definability and control of atypical intersections used to axiomatize the class C_B via sentences φ_V, but the authors rely on the Weak CIT (which they quote as a fact) because the full conjecture remains unresolved.

The authors explicitly state that CIT is still open and therefore use the weaker, provable form (Weak CIT) to carry out their arguments concerning rotund varieties and the axiomatization of strong embeddings.

References

Given an algebraic variety W\subseteq Kn over Q, there is a finite collection \mu(W) of proper algebraic subgroups of (K\times)n, satisfying the following property: If S is an atypical component of the intersection of W and a proper algebraic subgroup A\subseteq(K\times)n, i.e., \dimK S>\dimK W+\dimK A-n, then S is contained in some B\in\mu(W). Yet CIT is still open.

— Green points in the reals (2501.01176 - Zhang, 2 Jan 2025) in Section 2.2 (Facts in algebra), Conjecture [CIT] and subsequent sentence