Mall bundles and flat connections on Hopf manifolds

TL;DR

This paper introduces the concept of Mall bundles on Hopf manifolds, generalizes resonance notions from ODEs, and proves that non-resonant Mall bundles admit flat connections, leading to a linearization theorem for certain contractions.

Contribution

It defines resonant and non-resonant Mall bundles on Hopf manifolds and proves that non-resonant bundles admit flat holomorphic connections, extending classical linearization results.

Findings

Non-resonant Mall bundles always admit flat holomorphic connections.

A version of Poincare-Dulac linearization theorem is established for non-resonant contractions.

All non-resonant Hopf manifolds are shown to be linear.

Abstract

A Mall bundle on a Hopf manifold H is a holomorphic vector bundle whose pullback to the universal cover of H is trivial. We define resonant and non-resonant Mall bundles, generalizing the notion of the resonance in ODE, and prove that a non-resonant Mall bundle always admits a flat holomorphic connection. We use this observation to prove a version of Poincare-Dulac linearization theorem, showing that any non-resonant invertible holomorphic contraction of a complex space is linear in appropriate holomorphic coordinates. We define the notion of resonance in Hopf manifolds, and show that all non-resonant Hopf manifolds are linear; previously, this result was obtained by Kodaira using the Poincare-Dulac theorem.