Intersections and the B\'ezout Range: Abelian Varieties

TL;DR

This paper investigates intersection properties of subvarieties in simple abelian varieties, revealing that for most integers n, the intersections are zero-dimensional or empty, with implications for the Zilber–Pink conjecture.

Contribution

It establishes new results on the intersection behavior of subvarieties under multiplication in simple abelian varieties, extending classical intersection theory.

Findings

Intersections X ∩ [n]Y are zero-dimensional for all but finitely many n.

The union of these intersections is analytically dense in X.

For dim X + dim Y < dim A, intersections are empty for density-one set of n.

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.