Classifying spaces of degenerating mixed Hodge structures, IV: The fundamental diagram

TL;DR

This paper completes the construction of a fundamental diagram relating various compactifications of moduli spaces of mixed Hodge structures, aiding the understanding of degenerations and asymptotic behaviors in Hodge theory.

Contribution

It finalizes the fundamental diagram connecting different orbit spaces in mixed Hodge structures and explores their relations and applications to degeneration phenomena.

Findings

Complete construction of the fundamental diagram for mixed Hodge structures.

Amplification of relations among orbit spaces.

Application to asymptotic analysis of regulators and height pairings.

Abstract

We complete the construction of the fundamental diagram of various partial compactifications of the moduli spaces of mixed Hodge structures with polarized graded quotients. The diagram includes the space of nilpotent orbits, the space of SL(2)-orbits, and the space of Borel--Serre orbits. We give amplifications of this fundamental diagram, and amplify the relations of these spaces. We describe how this work is useful to understand asymptotic behaviors of Beilinson regulators and of local height parings in degeneration. We discuss "mild degenerations" in which regulators converge.