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On Special Subvarieties of the Universal Semi-abelian Scheme and Pink Conjectures

Kaiyuan Gu, Chenxin Huang

Abstract

The universal Poincaré torsor, or more generally the universal semi-abelian scheme, can be viewed as a mixed Shimura variety. We give a classification of special subvarieties of the universal semi-abelian scheme of arbitrary toric rank. Given this classification, we show that the Zilber-Pink conjecture for mixed Shimura varieties implies the Zilber-Pink conjecture for semi-abelian varieties, correcting an error in an unpublished manuscript of Pink. Moreover, we give a more reasonable reformulation of the Relative Manin-Mumford Conjecture for semi-abelian schemes.

On Special Subvarieties of the Universal Semi-abelian Scheme and Pink Conjectures

Abstract

The universal Poincaré torsor, or more generally the universal semi-abelian scheme, can be viewed as a mixed Shimura variety. We give a classification of special subvarieties of the universal semi-abelian scheme of arbitrary toric rank. Given this classification, we show that the Zilber-Pink conjecture for mixed Shimura varieties implies the Zilber-Pink conjecture for semi-abelian varieties, correcting an error in an unpublished manuscript of Pink. Moreover, we give a more reasonable reformulation of the Relative Manin-Mumford Conjecture for semi-abelian schemes.

Paper Structure

This paper contains 26 sections, 22 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Motivations of the Problem
  3. Main Theorems
  4. Reformulation of the Relative Manin-Mumford Conjecture
  5. Zilber-Pink for mixed Shimura varieties implies Zilber-Pink for semi-abelian varieties
  6. Structure of the Article
  7. Construction of Ribet Sections
  8. Ribet Sections as Special Subvarieties
  9. Review of Mixed Hodge Structures
  10. Pure Hodge Structures
  11. Mixed Hodge Structures
  12. Mixed Shimura Data of Ribet Sections
  13. Description of a Hodge Tensor
  14. Construction of the Mixed Shimura Data
  15. Classification of Special Subvarieties of the Poincaré Torsor
  16. ...and 11 more sections

Key Result

Theorem 1.3

For an irreducible subvariety $T\hookrightarrow\mathcal{P}^{\times}_g$, let $S=\Pi(T)\hookrightarrow {S_{g}}$. Then, $T$ is a special subvariety of $\mathcal{P}^{\times}_g$ if and only if the following holds:

Theorems & Definitions (61)

  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Definition 1.6
  • Theorem 1.7
  • Definition 1.8
  • Conjecture 1.9: Relative Manin-Mumford Conjecture for semi-abelian schemes
  • proof
  • Definition 2.4
  • ...and 51 more