Intersections and the B\'ezout Range: Abelian Varieties

Abstract

Given subvarieties $X, Y$ of a complex algebraic variety $S$ of complementary dimension, must they intersect? When $S$ is projective space, this is a consequence of the classical B\'ezout theorem, and an analogue for simple abelian varieties was established by Barth in 1968. Moreover, the moving lemma suggests that, after suitable translations, one may arrange for intersections of the expected dimension. In this work, we obtain variants for simple abelian varieties in the spirit of the completed Zilber--Pink philosophy. When $X$ and $Y$ have complementary dimension, we show that the intersections $X \cap [n]Y$ are zero-dimensional for all but finitely many integers $n$, and that these intersections collectively give rise to an analytically dense subset of $X$ as $n$ varies. We moreover control those $n$ for which $X \cap [n] Y$ has a positive dimensional component uniformly in $X, Y$ and $A$. When $\dim X + \dim Y < \dim A$, we show that $X \cap [n]Y = \varnothing$ for a set of integers $n$ of asymptotic density one, except in the presence of intersections at torsion points.