Counterexamples related to the Kobayashi Pseudodistance

TL;DR

This paper provides counterexamples showing that properties of the Kobayashi pseudodistance and related metrics do not necessarily transfer between base spaces, total spaces, or fibers in certain complex geometric contexts.

Contribution

It introduces unexpected counterexamples that challenge assumptions about the behavior of the Kobayashi pseudodistance in fiber bundles and coverings.

Findings

Vanishing of the Kobayashi pseudodistance on the base does not imply it on the total space.

Vanishing of the pseudodistance does not imply the vanishing of the Kobayashi Royden pseudometric.

The pseudodistance can vanish on the total space even if the fiber is hyperbolic.

Abstract

We present some unexpected examples related to the Kobayashi pseudodistance: For an unramified covering, the vanishing of the Kobayashi pseudodistance on the base does not imply the vanishing on the total space. The vanishing of the Kobayashi pseudodistance does not imply the vanishing of the Kobayashi Royden pseudo metric. Given a locally holomorphic trivial fiber bundle, the Kobayashi pseudodistance may vanish identically on the total space even if the fiber is hyperbolic.