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 \(\pi:D \to \Gamma \backslash D = S\) realize \(S\) as a quotient of \(D\), a homogeneous space for the action of a real algebraic group \(G\), by the action of \(\Gamma < G\), an arithmetic subgroup. Let \(S' \subseteq S\) be a special subvariety of \(S\) realized as \(\pi(D')\) for \(D' \subseteq D\) 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 \(S'\) is persistently likely meaning that whenever \(\zeta:S_1 \to S\) and \(\xi:S_1 \to S_2\) are maps of Shimura varieties (meaning regular maps of varieties induced by maps of the corresponding Shimura data) with \(\zeta\) finite, \(\dim \xi \zeta^{-1} X + \dim \xi \zeta^{-1} S' \geq \dim \xi S_1\). Then \(X \cap \bigcup_{g \in G, \pi(g D') \text{ is special }} \pi(g D')\) is dense in \(X\) for the Euclidean topology.