Formality and hard Lefschetz property of aspherical manifolds

For a Lie group $G=\R^{n}\ltimes_{\phi}\R^{m}$ with the semi-simple action $\phi:\R^{n}\to {\rm Aut}(\R^{m})$, we show that if $\Gamma$ is a finite extension of a lattice of $G$ then $K(\Gamma, 1)$ is formal. Moreover we show that a compact symplectic aspherical manifold with the fundamental group $\Gamma$ satisfies the hard Lefschetz property. By those results we give many examples of formal solvmanifolds satisfying the hard Lefschetz property but not admitting K\"ahler structures.Comment: 14 pages to appear in Osaka J. Mat

Flat bundles and Hyper-Hodge decomposition on solvmanifolds

We study rank $1$ flat bundles over solvmanifolds whose cohomologies are non-trivial. By using Hodge theoretical properties for all topologically trivial rank $1$ flat bundles, we represent the structure theorem of K\"ahler solvmanifolds as extensions of Hasegawa's result and Benson-Gordon's result for nilmanifolds.Comment: To appear in Int. Math. Res. Not. IMR

DGA-Models of variations of mixed Hodge structures

We define objects over Morgan's mixed Hodge diagrams which will be algebraic models of unipotent variations of mixed hodge structures over K\"ahler manifolds. As an analogue of Hain-Zucker's equivalence between unipotent variations of mixed Hodge structures and mixed Hodge representations of the fundamental group with Hain's mixed hodge structure, we give an equivalence between the category of our VMHS-like objects and the category of mixed Hodge representations of the dual Lie algebra of Sullivan's minimal model with Morgan's mixed Hodge structure. By this result, we can put various (tannakian theoretical) non-abelian mixed Hodge structures on the category of our new objects like the taking fibers of variations of mixed Hodge structures at points. By certain modifications of the result, we also give models of non-unipotent variations of mixed Hodge structures.Comment: 30 page