Rational parallelism on complex manifolds

TL;DR

This paper investigates rational parallelisms on complex manifolds, demonstrating that unlike holomorphic parallelisms, some rational parallelisms on compact complex manifolds are not flat with respect to any complex Lie algebra structure.

Contribution

It introduces the concept of rational parallelisms on complex manifolds and provides examples that are not flat, expanding understanding beyond holomorphic cases.

Findings

Rational parallelisms can exist without flatness.

Not all rational parallelisms are associated with complex Lie algebra structures.

The paper provides explicit examples of non-flat rational parallelisms.

Abstract

A Theorem of Wang in [Wa] implies that any holomorphic parallelism on a compact complex manifold M is flat with respect to some complex Lie algebra structure whose dimension coincides with that of M. We study here rational parallelisms on complex manifolds. We exhibit rational parallelisms on compact complex manifolds which are not flat with respect to any complex Lie algebra structure.