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Periods of Real Biextensions

Richard Hain

TL;DR

The paper investigates real biextensions, i.e., real mixed Hodge structures that extend $R(0)$ by a weight-graded pair of pieces of weights $-1$ and $-2$, in unipotent families over smooth varieties. It proves a key non-degeneracy result: if such a unipotent real biextension has non-abelian monodromy (or abelian monodromy with nontrivial action on the fiber), the general fiber does not split, and thus the associated period map into $iC_\ ext{R}^{-1,-1}$ is nontrivial. A central technical achievement is an explicit, canonical formula for the period map, derived from the canonical path-torsor variation $\mathbb{Z}\pi_1(X;p,q)/W_{-3}$ and Chen’s iterated integrals, both in the projective and the general quasi-projective settings. The results provide a powerful tool for understanding the boundary behavior of normal functions and are applied to study the Ceresa cycle boundary behavior, highlighting the practical impact for questions in algebraic cycles and Hodge theory.

Abstract

A real biextension is a real mixed Hodge structure that is an extension of R(0) by a mixed Hodge structure with weights $-1$ and $-2$. A unipotent real biextension over an algebraic manifold is a variation of mixed Hodge structure over it, each of whose fibers is a real biextension and whose weight graded quotients are do not vary. We show that if a unipotent real biextension has non abelian monodromy, then its ``general fiber'' does not split. This result is a tool for investigating the boundary behaviour of normal functions and is applied in arXiv:2408.07809 to study the boundary behaviour of the normal function of the Ceresa cycle.

Periods of Real Biextensions

TL;DR

The paper investigates real biextensions, i.e., real mixed Hodge structures that extend by a weight-graded pair of pieces of weights and , in unipotent families over smooth varieties. It proves a key non-degeneracy result: if such a unipotent real biextension has non-abelian monodromy (or abelian monodromy with nontrivial action on the fiber), the general fiber does not split, and thus the associated period map into is nontrivial. A central technical achievement is an explicit, canonical formula for the period map, derived from the canonical path-torsor variation and Chen’s iterated integrals, both in the projective and the general quasi-projective settings. The results provide a powerful tool for understanding the boundary behavior of normal functions and are applied to study the Ceresa cycle boundary behavior, highlighting the practical impact for questions in algebraic cycles and Hodge theory.

Abstract

A real biextension is a real mixed Hodge structure that is an extension of R(0) by a mixed Hodge structure with weights and . A unipotent real biextension over an algebraic manifold is a variation of mixed Hodge structure over it, each of whose fibers is a real biextension and whose weight graded quotients are do not vary. We show that if a unipotent real biextension has non abelian monodromy, then its ``general fiber'' does not split. This result is a tool for investigating the boundary behaviour of normal functions and is applied in arXiv:2408.07809 to study the boundary behaviour of the normal function of the Ceresa cycle.
Paper Structure (26 sections, 20 theorems, 124 equations)

This paper contains 26 sections, 20 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. A note to the reader
  4. Conventions
  5. Moduli of biextensions
  6. How to compute the period of a real biextension
  7. Unipotent quotients of path torsors
  8. A de Rham theorem
  9. The Hodge biextension associated to a path torsor
  10. The projective case
  11. The general case
  12. The period of the MHS on $\mathbb R\pi_1(X;p,q)/W_{-3}$
  13. The projective case
  14. Computation of the sections over $K(2,0)$
  15. Computation of the sections over $K(1,1)$
  16. ...and 11 more sections

Key Result

Theorem 1

Suppose that $X$ is a quasi-projective manifold and that $\mathbb V$ is a unipotent real biextension over $X$. If either then there is a nowhere dense real analytic subvariety $\Sigma$ of $X$ such that the fiber $V_x$ over $x\in X$ splits if and only if $x\in\Sigma$.

Theorems & Definitions (34)

  • Theorem 1
  • Remark 1.1
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Remark 3.1
  • Lemma 4.1
  • Proposition 4.2
  • proof
  • Lemma 4.3
  • ...and 24 more