We consider a special class of compact complex nilmanifolds, which we call compact nilmanifolds with nilpotent complex structure. It is shown that if $\Gamma \backslash G$ is a compact nilmanifold with nilpotent complex structure, then the Dolbeault cohomology $H^{\ast ,\ast }_{\bar {\partial } }(\Gamma \backslash G)$ is canonically isomorphic to the $\bar {\partial }$ –cohomology $H^{\ast ,\ast }_{\bar {\partial } }(\mathfrak {g}^{\mathbb {C} })$ of the bigraded complex $(\Lambda ^{\ast ,\ast } (\mathfrak {g} ^{\mathbb {C} })^{\ast }, \bar {\partial } )$ of complex valued left invariant differential forms on the nilpotent Lie group $G$ .

We exhibit a variational approach to study the magnetic flow associated with a Killing magnetic field in dimension 3. In this context, the solutions of the Lorentz force equation are viewed as Kirchhoff elastic rods and conversely. This provides an amazing connection between two apparently unrelated physical models and, in particular, it ties the classical elastic theory with the Hall effect. Then, these magnetic flows can be regarded as vortex filament flows within the localized induction approximation. The Hasimoto transformation can be used to see the magnetic trajectories as solutions of the cubic nonlinear Schrödinger equation showing the solitonic nature of those.