Complex connections with trivial holonomy

TL;DR

This paper investigates special complex connections with trivial holonomy on almost complex manifolds, focusing on their torsion types and natural occurrences in Lie groups with specific complex structures.

Contribution

It characterizes complex connections with trivial holonomy and specific torsion types, highlighting their natural emergence in Lie groups with bi-invariant and abelian complex structures.

Findings

Connections with torsion of type (2,0) or (1,1) are studied in detail.

Such connections naturally appear in Lie groups with special complex structures.

The paper provides a framework for understanding these connections in geometric contexts.

Abstract

Given an almost complex manifold (M, J), we study complex connections with trivial holonomy and such that the corresponding torsion is either of type (2,0) or of type (1,1) with respect to J. Such connections arise naturally when considering Lie groups, and quotients by discrete subgroups, equipped with bi-invariant and abelian complex structures.