MAXIMAL NILPOTENT COMPLEX STRUCTURES

Let the pair (g, J ) be a nilpotent Lie algebra 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 [6], we prove that 2 ≤ v ( J ) ≤ 3 for (g, J ) when v (g) = 2, where v (g) is the step of g and v ( J ) is the unique smallest integer such that a( J ) v ( J ) = g as in the [6, Def. 1, 8]. When v (g) = 3, for arbitrary n ≥ 3, there exists a pair (g, J ) such that v ( J ) = dim C g = n , for which we call the J in the pair (g, J ), satisfying v ( J ) = dim C 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 (g, J ), where v (g) = 3 and J is a MaxN complex structure.