Invariant and non-invariant almost complex structures on compact quotients of Lie groups
Lorenzo Sillari, Adriano Tomassini
TL;DR
This paper surveys invariant vs non-invariant almost complex structures on compact quotients of Lie groups, with a focus on the canonical bundle and Kodaira dimension. It reviews existence and classification results for complex and almost complex structures on Lie algebras, analyzes invariant cohomologies, and discusses how invariant structures often yield trivial canonical bundles while non-invariant ones can have more varied Kodaira dimensions. The authors provide new computations of Kodaira dimensions for both invariant and non-invariant structures, and construct explicit non-invariant examples on the Iwasawa manifold that illustrate these phenomena. The work clarifies the distinct geometric and cohomological behaviors of invariant versus non-invariant structures in the context of nilmanifolds and solvmanifolds, with implications for deformation theory and holomorphic trivializations.
Abstract
In this paper we briefly survey the classical problem of understanding which Lie algebras admit a complex structure, put in the broader perspective of almost complex structures with special properties. We focus on the different behavior of invariant and non-invariant structures, with a special attention to their canonical bundle and Kodaira dimension. We provide new examples of computations of Kodaira dimension of invariant and non-invariant structures.