A hyperbolic Ax-Lindemann theorem in the cocompact case

This is a joint work with Emmanuel Ullmo.

This work is motivated by J.Pila's strategy to prove the Andre-Oort conjecture. One ingredient in the strategy is the following

conjecture:

Let S be a Shimura variety uniformised by a symmetric space X.

Let V be an algebraic subvariety of S. Maximal algebraic subvarieties of the preimage of V in X are precisely the

components of the preimages of weakly special subvarieties contained in V.

We will explain the proof of this conjecture in the case where S is compact.