Given subvarieties X, Y of a complex algebraic variety S of complementary dimension, must they intersect?

For projective space this follows from the classical Bézout theorem, and an analogue for simple abelian varieties was established by Barth. The moving lemma further suggests that, after suitable translations, one may even ensure intersections of the expected dimension.

I will present new refinements in the abelian setting and discuss extensions to Shimura varieties, framed within the ''completed Zilber–Pink philosophy.'' (Joint work with D. Urbanik)