Harmonic metrics on unipotent bundles over quasi-compact Kaehler manifolds

TL;DR

This paper develops a method to construct and analyze harmonic metrics on unipotent bundles over quasi-compact Kähler manifolds, extending Schmid's Nilpotent Orbit Theorem to this setting with controlled asymptotics.

Contribution

It introduces a novel approach to harmonic metrics on unipotent bundles, providing existence, uniqueness, and canonical asymptotic behavior near divisors.

Findings

Constructed harmonic metrics with controlled asymptotics

Proved uniqueness of the harmonic metric up to isometry

Extended Nilpotent Orbit Theorem to unipotent bundles

Abstract

In this note, we propose an approach to the study of the analogue for unipotent harmonic bundles of Schmid's Nilpotent Orbit Theorem. Using this approach, we construct harmonic metrics on unipotent bundles over quasi-compact K\"ahler manifolds with carefully controlled asymptotics near the compactifying divisor; such a metric is unique up to some isometry. Such an asymptotic behavior is canonical in some sense.