Quasi-K\"ahler manifolds with trivial Chern Holonomy

TL;DR

This paper investigates compact quasi-K"ahler manifolds with trivial Chern holonomy, showing they are nilmanifolds and characterizing when such structures can be tamed by symplectic forms.

Contribution

It provides a classification of compact quasi-K"ahler Chern-flat manifolds as nilmanifolds and characterizes when these structures are compatible with symplectic forms.

Findings

Compact quasi-K"ahler Chern-flat manifolds are nilmanifolds.

Such structures can be tamed by a symplectic form only on flat tori.

Partial classification results for these manifolds.

Abstract

In this paper we study almost complex manifolds admitting a quasi-K\"ahler Chern-flat metric (Chern-flat means that the holonomy of the Chern connection is trivial). We prove that in the compact case such manifolds are all nilmanifolds. Some partial classification results are established and we prove that a quasi-K\"ahler Chern-flat structure can be tamed by a symplectic form if and only if the ambient space is isomorphic to a flat torus.