Periods of Real Biextensions

TL;DR

This paper investigates the properties of unipotent real biextensions in mixed Hodge structures, showing that non-abelian monodromy prevents splitting of the general fiber, with applications to boundary behavior of normal functions.

Contribution

It establishes a new link between non-abelian monodromy and the non-splitting of fibers in unipotent real biextensions, advancing understanding of their boundary behavior.

Findings

Non-abelian monodromy implies non-splitting of the general fiber.

Provides a tool for studying boundary behavior of normal functions.

Applied to the Ceresa cycle in subsequent work.

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.