| Résume | The Zilber-Pink conjectures predict that if S is a special variety, X ⊆ S is an irreducible subvariety of S which is not contained in a proper special subvariety, then the union of the unlikely intersections of X with special subvarieties of S is not Zariski dense in X, where here, an intersection between subvarieties X and Y of S is unlikely if dim X + dim Y < dim S. To make this precise, we need to specify what is meant by “special subvariety”. We will do so through the theory of definable complex quotient spaces, modeled on those introduced by Bakker, Klingler, and Tsimerman. Using this formalism we will prove a complement to the Zilber-Pink conjecture to the effect that under some natural geometric conditions likely intersections will be Zariski dense in X (joint work with Sebastian Eterović) and in the other direction that a function field version of the Zilber-Pink conjecture holds effectively (joint work with Jonathan Pila). |
