Maximal nilpotent complex structures

Abstract

Let the pair $(\mathfrak{g},J)$ be a nilpotent Lie algebra $\mathfrak{g}$ (NLA for short) endowed with a nilpotent complex structure $J$. In this paper, motivated by a question in the work of Cordero, Fernández, Gray and Ugarte, we prove that $2\le \nu (J)\le 3$ for $(\mathfrak{g},J)$ when $\nu (\mathfrak{g})=2$, where $\nu (\mathfrak{g})$ is the step of $\mathfrak{g}$ and $\nu (J)$ is the unique smallest integer such that $\mathfrak{a}(J{)}_{\nu (J)}=\mathfrak{g}$ as in Definition 1 and 8 of the paper by Cordero, Fernández, Gray and Ugarte. When $\nu (\mathfrak{g})=3$, for arbitrary $n\ge 3$, there exists a pair $(\mathfrak{g},J)$ such that $\nu (J)={\dim }_{\mathbb{C}}\mathfrak{g}=n$, for which we call the $J$ in the pair $(\mathfrak{g},J)$, satisfying $\nu (J)={\dim }_{\mathbb{C}}\mathfrak{g}=n$, a maximal nilpotent (MaxN for short) complex structure. The algebraic dimension of a nilmanifold endowed with a left invariant MaxN complex structure is discussed. Furthermore, a structure theorem is proved for the pair $(\mathfrak{g},J)$, where $\nu (\mathfrak{g})=3$ and $J$ is a MaxN complex structure.