Complex structures on two-step nilpotent Lie groups

Abstract: We give a characterization of the $2$-step nilpotent Lie algebras whose corresponding Lie groups admit a left invariant complex structure. This is done by considering separately the cases when the complex structure is 2-step or 3-step nilpotent in the sense of [Cordero-Fern\'andez-Gray-Ugarte, 2000]. We also study the Hermitian geometry 2-step nilpotent Lie groups. We show that if a left invariant Hermitian metric on such Lie group is pluriclosed, then the corresponding complex structure is 2-step nilpotent. Moreover, we obtain a necessary and sufficient condition for such a metric to be pluriclosed in case the complex structure is abelian. This allows us to show that pluriclosed metrics on nilpotent Lie algebras with one dimensional commutator ideal can only occur on trivial central extensions of the $3$-dimensional Heisenberg Lie algebra. We show that certain Hermitian nilmanifolds constructed from compact semisimple irreducible Hermitian symmetric pairs are pluriclosed with respect to a canonically defined abelian complex structure. Finally, we give necessary and sufficient conditions for a naturally reductive Riemannian metric on a nilmanifold to be Hermitian with respect to an abelian complex structure. We prove the analogue of this result in the hypercomplex case, thereby obtaining a distinguished family of hyper-K\"ahler with torsion metrics.