On the geometry and cohomology of almost abelian solvmanifolds

Abstract

We study left-invariant generalized complex structures on almost abelian Lie groups $G_A = \mathbb{R} \ltimes_{e^{tA}} \mathbb{R}^{d}$, and on the associated solvmanifolds. Our starting point is that the $\mathbf{i}$-eigenspace $\mathfrak{L}$ of such a structure is itself an almost abelian complex Lie algebra, a fact that governs everything that follows. When the structure matrix $A$ is diagonalizable over $\mathbb R$, we classify the types that occur in terms of the spectrum of $A$: they are prescribed by pairs of identical eigenvalues, pairs of opposite eigenvalues, and one remaining eigenvalue, the extremal types recovering the known classification of left-invariant complex and symplectic structures. We characterize the generalized Calabi--Yau case by a single linear condition on the eigenvalues, exhibit groups admitting only structures of intermediate type, and prove that admissible types come in adjacent pairs. Beyond the diagonalizable case we give an upper bound for the type in terms of the Jordan data of $A$, in dimension $6$, classify (non-extremal) left-invariant generalized complex structures, the first dimension in which these can occur. Finally, we compute the generalized Dolbeault cohomologies of the classified structures: it is given by a closed counting formula over sub-multisets of that spectrum, together with a duality in the grading.