We study the interplay between following types of special non-Kähler Hermitian metrics on compact complex manifolds (locally conformally Kähler, k-Gauduchon, balanced, and locally balanced) prove that a Kähler nilmanifold carrying balanced or left-invariant k-Gauduchon metric is necessarily torus. Combined with main result in [FV16], this leads to fact 2-step endowed whichever two metrics—balan...

We develop deformation theory for abelian invariant complex structures on a nilmanifold, and prove that in this case the invariance property is preserved by the Kuranishi process. A purely algebraic condition characterizes the deformations leading again to abelian structures, and we prove that such deformations are unobstructed. Various examples illustrate the resulting theory, and the behavior...

We continue the study of the distribution of closed geodesics on nilmanifolds Γ\N , constructed from a simply connected 2-step nilpotent Lie group N with a left invariant metric and a lattice Γ in N . We consider a Lie group N with associated 2-step nilpotent Lie algebra N constructed from an irreducible representation of a compact semisimple Lie algebra g0 on a real finite dimensional vector s...

We study the Anomaly flow on $2$-step nilmanifolds with respect to any Hermitian connection in Gauduchon line. In case of flat holomorphic bundle, general solution is given for initial invariant metric. The solutions depend two constants $K_1$ and $K_2$, we qualitative behaviour terms their signs, as well convergence Gromov-Hausdorff topology. sign related conformal introduced by Fu, Wang Wu. n...