We investigate some properties of complex structures on Lie algebras. In particular, we focus on nilpotent complex structures that are characterised by suitable J-invariant ascending or descending central series, $\mathfrak {d}^{\,j}$ and $\mathfrak {d}_j$ , respectively. We introduce a new descending series $\mathfrak {p}_j$ and use it to prove a new characterisation of nilpotent complex structures. We also examine whether nilpotent complex structures on stratified Lie algebras preserve the strata. We find that there exists a J-invariant stratification on a step $2$ nilpotent Lie algebra with a complex structure.

Given a separable Banach space E, we construct an extremely non-complex Banach space (i.e. a space satisfying that ‖ Id + T2 ‖ = 1 + ‖ T2 ‖ for every bounded linear operator T on it) whose dual contains E* as an L-summand. We also study surjective isometries on extremely non-complex Banach spaces and construct an example of a real Banach space whose group of surjective isometries reduces to ±Id, but the group of surjective isometries of its dual contains the group of isometries of a separable infinite-dimensional Hilbert space as a subgroup.

For an integer $n\,\ge \,3$, let ${{M}_{n}}$ be the moduli space of spatial polygons with $n$ edges. We consider the case of odd $n$. Then ${{M}_{n}}$ is a Fano manifold of complex dimension $n\,-\,3$. Let ${{\Theta }_{{{M}_{n}}}}$ be the sheaf of germs of holomorphic sections of the tangent bundle $T{{M}_{n}}$. In this paper, we prove ${{H}^{q}}\left( {{M}_{n}},\,{{\Theta }_{{{M}_{n}}}} \right)\,=\,0$ for all $q\,\ge \,0$ and all odd $n$. In particular, we see that the moduli space of deformations of the complex structure on ${{M}_{n}}$ consists of a point. Thus the complex structure on ${{M}_{n}}$ is locally rigid.