Complex nilmanifolds and Kähler-like connections

In this note, we analyze the question of when will a complex nilmanifold have Kähler-like Strominger (also known as Bismut), Chern, or Riemannian connection, in the sense that the curvature of the connection obeys all the symmetries of that of a Kähler metric. We give a classification in the second case and a partial description in the first and the third case. It would be interesting to understand these questions for all Lie–Hermitian manifolds, namely, Lie groups equipped with a left invariant complex structure and a compatible left invariant metric.