Intersections and the Bézout 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 Bézout 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.

• The paper presents a uniform finiteness result showing that non-proper intersections occur for only finitely many endomorphisms when subvarieties lie in the Bézout range.
• It employs advanced methods from Hodge theory, o-minimality, and analytic uniformization to analyze intersection density and equidistribution of translated subvarieties.
• The results extend classical Bézout theorems using the Zilber–Pink framework, offering quantitative tools for unexpected intersections in arithmetic geometry.

Intersections and the Bézout Range in Simple Abelian Varieties

Introduction

The paper "Intersections and the Bézout Range: Abelian Varieties" (2604.02186) presents a comprehensive study of intersection theory for irreducible subvarieties within simple abelian varieties, focusing on the dimensional thresholds set by the classical Bézout theorem and extending the framework to analogues inspired by the Zilber–Pink conjecture. The work investigates the nature, distribution, and uniformity of intersections when acting on subvarieties via group endomorphisms, rather than just translations. The results articulate both strong qualitative and quantitative descriptions, especially concerning the density, dimensionality, and atypicality of intersections, and establish new uniformity claims rooted in recent advances in Hodge theory and o-minimality.

Classical and Modern Bézout Statements

The classical Bézout theorem asserts that in projective space, two subvarieties whose dimensions sum to at least the ambient variety's dimension must intersect nontrivially. Its analogue in simple abelian varieties was established by Barth, formalizing that for irreducible subvarieties $X, Y \subset A$ (with $A$ simple abelian), $\dim X + \dim Y \geq \dim A$ implies $X\cap Y \neq \emptyset$. However, proper intersection is not guaranteed, especially due to an abundance of automorphisms and self-intersections in abelian varieties.

Crucially, the classical moving lemma in intersection theory is recast: instead of considering arbitrary translations, the paper studies intersections after acting on $Y$ by group endomorphisms, particularly $[n]:A \to A$ for integers $n$, uncovering more nuanced and chaotic intersection behaviors.

Bézout Range: Expected versus Unexpected Intersections

For subvarieties $X, Y$ in the "Bézout range" ($\dim X + \dim Y \geq \dim A$), the paper establishes that for all but finitely many $n\in\mathbb{Z}$, $A$0 and $A$1 intersect properly, i.e., their intersection is of the expected dimension. This phenomenon is rigorously analyzed via the geometric Zilber–Pink framework, providing a finite uniform bound (dependent on degrees and polarizations) for exceptional $A$2, independent of the choice of $A$3, and $A$4.

Moreover, the union of these expected intersections over an infinite set of $A$5 is analytically dense in $A$6, providing a strong equidistribution result. The proof utilizes techniques from VMHS, equidistribution in the tangent bundle, and sophisticated applications of analytic uniformization and monodromy groups.

Uniformity Theorem

A significant claim is the existence of a uniform constant $A$7 (determined by $A$8, $A$9, polarization degree, and dimension of $\dim X + \dim Y \geq \dim A$0) such that non-proper intersections occur for at most $\dim X + \dim Y \geq \dim A$1 endomorphisms, a result extracted via generalized geometric Zilber–Pink theorems for families over moduli spaces. This uniformity is not previously explicit in literature and provides critical control for applications in arithmetic geometry and unlikely intersection problems.

Outside the Bézout Range: Density and Exceptional Dynamics

When $\dim X + \dim Y \geq \dim A$2, intersections are "unexpected". The paper demonstrates that, except for the presence of torsion points, for a density $\dim X + \dim Y \geq \dim A$3 subset of integers $\dim X + \dim Y \geq \dim A$4, $\dim X + \dim Y \geq \dim A$5. Intersections at torsion points can persist for positive density subsets of $\dim X + \dim Y \geq \dim A$6, precisely characterized by modular congruences arising from subgroup generation in $\dim X + \dim Y \geq \dim A$7.

Furthermore, the count of non-torsion, nonempty intersections is bounded above by $\dim X + \dim Y \geq \dim A$8 for any $\dim X + \dim Y \geq \dim A$9, and recent advances allow this to be sharpened to $X\cap Y \neq \emptyset$0 for some $X\cap Y \neq \emptyset$1, building on Wilkie's conjecture. The proofs employ o-minimality (e.g., $X\cap Y \neq \emptyset$2), Pila–Zannier strategy, and functional transcendence from Ax–Schanuel.

Geometric Zilber–Pink Uniformity for Families

Extending the results to families of abelian varieties, the paper proves a uniform geometric Zilber–Pink theorem for VMHS, asserting that maximal atypical intersections (i.e., those whose codimension in $X\cap Y \neq \emptyset$3 is less than their codimension in the ambient variety) belong to finitely many algebraic families. This directly implies uniform bounds in intersection properties across moduli spaces, underpinning the explicit uniform bounds in earlier theorems and facilitating further generalization to Shimura varieties and other contexts.

Strong Results and Claims

• Finiteness: There are only finitely many $X\cap Y \neq \emptyset$4 such that $X\cap Y \neq \emptyset$5 and $X\cap Y \neq \emptyset$6 fail to meet properly inside a simple abelian variety, uniformly, provided $X\cap Y \neq \emptyset$7.
• Density: For any infinite sequence of endomorphisms, the union of proper intersections is analytically dense in $X\cap Y \neq \emptyset$8.
• Asymptotic Non-Intersection: Outside the Bézout range, $X\cap Y \neq \emptyset$9 for asymptotic density $Y$0 subset of $Y$1, unless torsion congruence conditions are satisfied.
• Uniformity: Uniform geometric Zilber–Pink yields effective, algebraically bounded families of maximal atypical intersections in families of abelian varieties.

Implications and Future Directions

The results sharpen the theoretical understanding of intersection behaviors under endomorphism actions in abelian varieties. The uniformity and density claims provide foundational tools for arithmetic applications, such as Manin–Mumford, Mordell–Lang, and unlikely intersection problems in modern Diophantine geometry. The explicit bounds and analytic density are crucial for quantitative results and algorithmic applications in arithmetic geometry.

The extensions to VMHS and families pave the way for analogous results in Shimura varieties and other moduli spaces, which are planned for subsequent work. The o-minimal and functional transcendence methods underscore a robust bridge between geometric, analytic, and arithmetic approaches. The paper's techniques and uniformity assertions are likely to stimulate further developments in effective unlikely intersection theory and arithmetic dynamics.

Conclusion

The paper systematically analyzes intersection properties of subvarieties of simple abelian varieties within and outside the Bézout range, yielding uniform, finite, and density results via advanced intersection theory, geometric Zilber–Pink, and arithmetic geometry. The implications are substantial for both theoretical trajectories and practical applications in the study of unlikely intersections, arithmetic moduli, and algebraic dynamics, with future prospects in higher-dimensional Shimura varieties and mixed Hodge theoretic settings.