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On Some Systems of Equations in Abelian Varieties

Francesco Gallinaro

TL;DR

This work proves a case of Abelian Exponential-Algebraic Closedness (AEAC) for complex abelian varieties, showing that every free and rotund subvariety $V \subseteq \mathbb{C}^g \times A$ intersects the graph $\Gamma_{\exp_A}$ of the abelian exponential map. The author develops a self-contained framework combining singular homology, transversality of intersections, and o-minimal definability to connect algebraic geometry with exponential-algebraic solvability; in particular, for varieties of the form $V = L \times W$ with $L$ linear and $W$ algebraic and satisfying freeness/rotundity, the intersection $V \cap \Gamma_{\exp_A}$ is guaranteed, and this leads to Zariski-density results for $\exp_A(L) \cap W$ in $W$ under suitable hypotheses. The main result extends prior exponential-case techniques to abelian varieties, using homological duality and open-ness arguments for the $\delta$-map rather than tropical geometry, and it reinforces the model-theoretic program linking AEAC to quasiminimality of structures on $\mathbb{C}$ with $\exp_A$. Together with the detailed geometric preliminaries and an explicit example, the paper provides a self-contained approach to solving a broad class of equations involving polynomials and the abelian exponential map. The findings contribute to the understanding of the interaction between algebraic varieties and exponential maps in the complex setting, with potential implications for quasiminimality and related model-theoretic objectives.

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.

On Some Systems of Equations in Abelian Varieties

TL;DR

This work proves a case of Abelian Exponential-Algebraic Closedness (AEAC) for complex abelian varieties, showing that every free and rotund subvariety intersects the graph of the abelian exponential map. The author develops a self-contained framework combining singular homology, transversality of intersections, and o-minimal definability to connect algebraic geometry with exponential-algebraic solvability; in particular, for varieties of the form with linear and algebraic and satisfying freeness/rotundity, the intersection is guaranteed, and this leads to Zariski-density results for in under suitable hypotheses. The main result extends prior exponential-case techniques to abelian varieties, using homological duality and open-ness arguments for the -map rather than tropical geometry, and it reinforces the model-theoretic program linking AEAC to quasiminimality of structures on with . Together with the detailed geometric preliminaries and an explicit example, the paper provides a self-contained approach to solving a broad class of equations involving polynomials and the abelian exponential map. The findings contribute to the understanding of the interaction between algebraic varieties and exponential maps in the complex setting, with potential implications for quasiminimality and related model-theoretic objectives.

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 which split as the product of a linear subspace of the Lie algebra of 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.
Paper Structure (10 sections, 23 theorems, 42 equations, 1 figure)

This paper contains 10 sections, 23 theorems, 42 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Geometric Preliminaries
  3. Notation and Conventions
  4. Homology and Cohomology
  5. Definability and O-Minimality
  6. Abelian Exponential-Algebraic Closedness
  7. The Conjecture
  8. Consequences of Freeness and Rotundity
  9. Abelian E.A.C. for Varieties of the form $L \times W$
  10. An Example

Key Result

Theorem 1

Let $A$ be a complex abelian variety of dimension $g$ with exponential map $\textnormal{exp}_A:\mathbb{C}^g \rightarrow A$. Let $L \leq \mathbb{C}^g$ be a linear subspace and $W \subseteq A$ an algebraic variety such that the variety $L \times W$ is free and rotund. Then $L \times W \cap \Gamma_{\te

Figures (1)

  • Figure 1: As in the example, let $\Lambda_1=\mathbb{Z}+i\sqrt{2}\mathbb{Z}$ and $\Lambda_2=\mathbb{Z}+i \sqrt{5}\mathbb{Z}$. Given a point $z=x+iy \in \mathbb{C}$, the set of points of the form $(z,z+\lambda)$ for $\lambda \in \Lambda_2$ is dense in the set of points $(z, \textnormal{Re}(z)+y)$ for $y \in \mathbb{R}$. This is shown in this figure: for a fixed $z \in \mathbb{C}/\Lambda_1$ on the left, we see infinitely many determinations on the right, filling the vertical line.

Theorems & Definitions (57)

  • Definition 1.1
  • Conjecture 1.2: Abelian Exponential-Algebraic Closedness, see BK18
  • Theorem : Theorem \ref{['abmain']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • ...and 47 more