On the variety of Lie algebras endowed with complex structures: degenerations and deformations

We study the space of Lie algebras equipped with left-invariant complex structures, ${\mathcal{L}}_{J_{cn}}({\mathbb{R}}^{2n})$, with particular attention to their degenerations and deformations. To this end, we identify certain invariants that remain well-behaved under degenerations while preserving the complex structure. These concepts are then applied to the four-dimensional case. Additionally, we explore applications to the study of left-invariant Hermitian structures on Lie groups, and we discuss some aspects of the deformation theory within ${\mathcal{L}}_{J_{cn}}({\mathbb{R}}^{2n})$.