Hyperbolic Ax–Lindemann conjecture for Shimura varieties

Let $S$ be a Shimura variety and let ${\mathcal X}$ be a complex realisation of a connected component $X^+$. Let

$$\pi\colon {\mathcal X}\longrightarrow S$$

be the uniformisation map, and let $V$ be an algebraic subvariety of $S$. An algebraic subvariety of ${\mathcal X}$ is an irreducible analytic component of the intersection of ${\mathcal X}$ with a closed algebraic subvariety of an ambient complex algebraic variety. Hyperbolic Ax–Lindemann conjecture. Maximal algebraic subvarieties of $\pi^{-1}V$ are weakly special subvarieties. This is the functional, hyperbolic analogue of the classical Ax–Lindemann theorem and is intended to support Pila's strategy for the André–Oort conjecture. The paper proves this statement in the cocompact case, so the conjecture as stated here is solved in the setting addressed by the paper.