On the generalized cohomologies of almost abelian solvmanifolds

Abstract

Generalized complex structures give rise to generalized Dolbeault, Bott--Chern and Aeppli cohomologies. We study these cohomologies for invariant generalized complex structures on almost abelian Lie algebras ${\mathfrak{g}}_A=\mathbb{R}{e}_0{⋉}_A\mathfrak{h}$, where $\mathfrak{h}$ is an abelian ideal of codimension one, and on almost abelian solvmanifolds $\Gamma \backslash G_A$. We first show that the invariant generalized Dolbeault cohomology can be identified with the Chevalley--Eilenberg cohomology of the complex Lie algebra $\mathcal{L}$, the $+\mathbf{i}$-eigenspace of the structure, with coefficients in a character determined by the canonical line. We prove that $\mathcal{L}$ is itself an almost abelian Lie algebra; thus the cohomology reduces to the kernel and cokernel of an explicit derivation. Assuming that $A$ is diagonalizable over $\mathbb{R}$, we obtain a combinatorial formula for the cohomology dimensions and compare with compact quotients. Other comparison results are discussed, noting that the cohomology of invariant forms always injects into that of $\Gamma \backslash G_A$.