On Some Systems of Equations in Abelian Varieties
Abstract
We solve a case of the Abelian ExponentialAlgebraic 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 BaysKirby, 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 ominimality (definability of Hausdorff limits), hence we have tried to give a selfcontained exposition.
 Publication:

arXiv eprints
 Pub Date:
 June 2022
 DOI:
 10.48550/arXiv.2206.14074
 arXiv:
 arXiv:2206.14074
 Bibcode:
 2022arXiv220614074G
 Keywords:

 Mathematics  Logic;
 Mathematics  Algebraic Geometry;
 Mathematics  Complex Variables
 EPrint:
 41 pages, 3 figures