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Attractors are not algebraic

Yeuk Hay Joshua Lam, Arnav Tripathy

Abstract

The Attractor Conjecture for Calabi-Yau moduli spaces predicts the algebraicity of the moduli values of certain isolated points picked out by Hodge-theoretic conditions. We provide a family of counterexamples to the Attractor Conjecture in all suitably high, odd dimensions conditional on the Zilber-Pink conjecture.

Attractors are not algebraic

Abstract

The Attractor Conjecture for Calabi-Yau moduli spaces predicts the algebraicity of the moduli values of certain isolated points picked out by Hodge-theoretic conditions. We provide a family of counterexamples to the Attractor Conjecture in all suitably high, odd dimensions conditional on the Zilber-Pink conjecture.

Paper Structure

This paper contains 20 sections, 20 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Statement of results
  3. History of the problem and related works
  4. Outline of proof of Theorem \ref{['thm:main']}
  5. Outline of the rest of the paper
  6. Notations and conventions
  7. The Attractor Conjecture
  8. Dolgachev Calabi-Yau varieties
  9. Defining the Calabi-Yaus
  10. Hodge structures of Dolgachev Calabi-Yaus
  11. The attractor condition for Dolgachev Calabi-Yau varieties
  12. Algebraicity of the associated curve
  13. Attractors are dense
  14. Reduction to Shimura theory
  15. Algebraic attractors split off CM abelian varieties
  16. ...and 5 more sections

Key Result

Theorem 1.1.3

Under the Zilber-Pink conjecture, the analogue of Conjecture conj:attractor for Calabi-Yau varieties of arbitrary dimension is false. More precisely, there exist attractor Calabi-Yau varieties in all odd dimensions except 1, 3, 5 and 9 which are not defined over ${\overline{\mathbb{Q}}}$.

Theorems & Definitions (59)

  • Conjecture 1.1.1: Moore
  • Definition 1.1.2
  • Theorem 1.1.3
  • Conjecture 1.1.4
  • Definition 2.0.1
  • Remark 2.0.2
  • Conjecture 2.0.3: moore for the case of threefolds
  • Definition 3.1.1
  • Proposition 3.1.2
  • Definition 3.1.4
  • ...and 49 more