On the generalized geometry of almost abelian solvmanifolds

Abstract

We study left-invariant generalized complex structures on almost abelian Lie groups $G_A$ with Lie algebra $\mathfrak{g}_A=\mathbb{R} e_0\ltimes_A\mathfrak{h}$, where $\mathfrak{h}$ is an abelian ideal of codimension one, and on their compact quotients. First, when $A$ is diagonalizable over $\mathbb{R}$, we characterize all admissible types by pairings of its eigenvalues and characterize the structures admitting a closed invariant pure-spinor generator. For general $A$, we obtain Jordan-theoretic type bounds and a construction using a complex quotient and a symplectic ideal. Finally, in dimension six, we establish nonexistence results and give explicit intermediate-type constructions.