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Complex nilmanifolds and K\"ahler-like connections
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In this note, we analyze the question of when will a complex nilmanifold have K\"ahler-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 K\"ahler metric. We give a classification in the first two cases and a partial description in 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.
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On Strominger K\"ahler-like manifolds with degenerate torsion
Every complete non-Kähler SKL threefold has universal cover either a product of two Sasakian 3-manifolds or a non-Kähler SKL surface times a Kähler curve, and degenerate-torsion SKL manifolds split off a Kähler factor.
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