On Some Systems of Equations in Abelian Varieties

TL;DR

This paper proves a special case of the Abelian Exponential-Algebraic Closedness Conjecture, demonstrating solvability conditions for certain systems of equations in abelian varieties using techniques from topology, homology, and o-minimality.

Contribution

It establishes the conjecture for subvarieties of the tangent bundle of abelian varieties that split as a product, extending previous work and providing a self-contained proof.

Findings

Confirmed the conjecture for specific subvarieties of tangent bundles.

Connected the conjecture's validity to quasiminimality of complex structures.

Applied interdisciplinary techniques from topology, homology, and o-minimality.

Abstract

We solve a case of the Abelian Exponential-Algebraic Closedness Conjecture, a conjecture due to Bays and Kirby, building on work of Zilber, which predicts sufficient conditions for systems of equations involving algebraic operations and the exponential map of an abelian variety to be solvable in the complex numbers. More precisely, we show that the conjecture holds for subvarieties of the tangent bundle of an abelian variety $A$ which split as the product of a linear subspace of the Lie algebra of $A$ and an algebraic variety. This is motivated by work of Zilber and of Bays-Kirby, which establishes that a positive answer to the conjecture would imply quasiminimality of certain structures on the complex numbers. Our proofs use various techniques from homology (duality between cup product and intersection), differential topology (transversality) and o-minimality (definability of Hausdorff limits), hence we have tried to give a self-contained exposition.