Abstract
Complex manifolds can be characterized as pairs $(M,J)$, where~$M$ is a differentiable manifold of even dimension and~$J$ is an integrable almost-complex structure on~$M$. Although the explicit construction of such~$J$'s has proven to be a difficult task, the problem can be slightly simplified when~$M$ is a nilmanifold and one restricts to the study of \emph{invariant} complex structures on it. Under these assumptions, one can work on the nilpotent Lie algebra~$\mathfrak{g}$ underlying~$M$ and focus on those~$J$'s defined on it. One can then distinguish two types of complex structures, depending on whether the center of~$\mathfrak g$ admits a $J$-invariant subspace~$\mathfrak a_J\neq\{0\}$ or not. In this talk, we will focus on the case~$\mathfrak a_J=\{0\}$ and provide the complete classification of pairs $(\mathfrak g, J)$ for~$\mathfrak g$ an $8$-dimensional nilpotent Lie algebra.
