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Ax-Schanuel for variations of mixed Hodge structures

Kenneth Chung Tak Chiu

Abstract

We give properties of the real-split retraction of the mixed weak Mumford-Tate domain and prove the Ax-Schanuel property of period mappings arising from variations of mixed Hodge structures. An ingredient in the proof is the definability of the mixed period mapping obtained by Bakker-Brunebarbe-Klingler-Tsimerman. In comparison with preceding results, in the point counting step, we count rational points on definable quotients instead.

Ax-Schanuel for variations of mixed Hodge structures

Abstract

We give properties of the real-split retraction of the mixed weak Mumford-Tate domain and prove the Ax-Schanuel property of period mappings arising from variations of mixed Hodge structures. An ingredient in the proof is the definability of the mixed period mapping obtained by Bakker-Brunebarbe-Klingler-Tsimerman. In comparison with preceding results, in the point counting step, we count rational points on definable quotients instead.

Paper Structure

This paper contains 20 sections, 35 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Statement of results
  4. Ideas of proof
  5. Acknowledgements
  6. Base cases of induction
  7. Retraction of the mixed weak Mumford-Tate domain
  8. Normality of algebraic stabilizer in algebraic monodromy group
  9. A temporary definable fundamental set $\Phi'$ for $\textbf{P}(\mathbb{Z})^+\backslash D$
  10. Normality of algebraic stabilizer in algebraic monodromy group
  11. Definable quotient $\overline{I}$
  12. Definable fundamental set $\Phi$ for $\textbf{P}(\mathbb{Z})^+\backslash D$
  13. Definable subset $I$ of $\textbf{P}(\mathop{\mathrm{\mathbb{C}}}\nolimits)$
  14. Definable quotient $\overline{I}$
  15. Heights and trichotomy
  16. ...and 5 more sections

Key Result

Theorem 1.2

If $\dim V-\dim U< \dim \widecheck{D},$ then $p_X(U)$ is contained in a proper weakly special subvariety.

Theorems & Definitions (71)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 61 more