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Likely intersections

We prove a general likely intersections theorem, a counterpart to the Zilber-Pink conjectures, under the assumption that the Ax-Schanuel property and some mild additional conditions are known to hold for a given category of complex quotient spaces definable in some fixed o-minimal expansion of the ordered field of real numbers. For an instance of our general result, consider the case of subvarieties of Shimura varieties. Let $S$ be a Shimura variety. Let $π:D \to Γ\backslash D = S$ realize $S$ as a quotient of $D$, a homogeneous space for the action of a real algebraic group $G$, by the action of $Γ< G$, an arithmetic subgroup. Let $S&#39; \subseteq S$ be a special subvariety of $S$ realized as $π(D&#39;)$ for $D&#39; \subseteq D$ a homogeneous space for an algebraic subgroup of $G$. Let $X \subseteq S$ be an irreducible subvariety of $S$ not contained in any proper weakly special subvariety of $S$. Assume that the intersection of $X$ with $S&#39;$ is persistently likely meaning that whenever $ζ:S_1 \to S$ and $ξ:S_1 \to S_2$ are maps of Shimura varieties (meaning regular maps of varieties induced by maps of the corresponding Shimura data) with $ζ$ finite, $\dim ξζ^{-1} X + \dim ξζ