Effective transcendental Zilber-Pink for variations of Hodge structures

Abstract

We prove function field versions of the Zilber-Pink conjectures for varieties supporting a variation of Hodge structures. A form of these results for Shimura varieties in the context of unlikely intersections is the following. Let $S$ be a connected pure Shimura variety with a fixed quasiprojective embedding. We show that there is an explicitly computable function $B$ of two natural number arguments so that for any field extension $K$ of the complex numbers and Hodge generic irreducible proper subvariety $X \subsetneq S_K$, the set of nonconstant points in the intersection of $X$ with the union of all special subvarieties of $X$ of dimension less than the codimension of $X$ in $S$ is contained in a proper subvariety of $X$ of degree bounded by $B(\operatorname{deg}(X),\dim(X))$. Our techniques are differential algebraic and rely on Ax-Schanuel functional transcendence theorems. We use these results to show that the differential equations associated with Shimura varieties give new examples of minimal, and sometimes, strongly minimal, types with trivial forking geometry but non-$\aleph_0$-categorical induced structure.