The variety of Lie algebras endowed with complex structures and their degenerations

The study of complex structures on Lie groups provides a natural bridge between algebra, geometry, and complex analysis. In this talk, we focus on Lie algebras endowed with left-invariant complex structures, and on how these structures behave under degenerations and deformations.

A central question is how to understand the possible “limits” of such structures and which features remain stable under these processes. To address this, we introduce certain invariants that are well adapted to degenerations while preserving the complex structure.

We illustrate these ideas in the four-dimensional case, where a more concrete picture can be obtained.