In his book "G-functions and Geometry", Y. André considers a curve $S$ over a number field $K$ supporting an abelian scheme $f:X\rightarrow S$ and proves, under certain conditions on the curve, that the points in $S(\overline{K})$ for which the fiber $X_s$ has "too many endomorphisms" will have Weil heights bounded, essentially, by the degree of the field of definition of these points.

These bounds

were recently employed by C. Daw and M. Orr to establish cases of the Zilber-Pink conjecture in $\mathcal{A}_g$.

In this talk we describe how one can get similar height bounds when one replaces the abelian scheme by a variation of Hodge structures arising from some smooth projective morphism $f:X\rightarrow S$ over a curve $S$, under certain conditions on the family $f:X\rightarrow S$ and assuming the Hodge conjecture holds for our spaces.