Complex connections with trivial holonomy
  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. 
A. Andrada, M.L. Barberis, I.G. Dotti
math.DG