Minimal models, formality, and hard Lefschetz properties of solvmanifolds with local systems

Abstract

Abstract. For a simply connected solvable Lie group G with a cocompact discrete subgroup Γ, we consider the space of differential forms on the solvmanifold G/Γ with values in certain flat bundle so that this space has a structure of a differential graded algebra(DGA). We construct Sullivan's minimal model of this DGA. This result is an extension of Nomizu's theorem for ordinary coefficients in the nilpotent case. By using this result, we refine Hasegawa's result of formality of nilmanifolds and Benson-Gordon's result of hard Lefschetz properties of nilmanifolds.

Massey products in mapping tori

“…6.5). This is a strengthening of the formality condition, since there exist formal, but not strongly formal spaces (see H. Kasuya's paper [6], §9, Example 1 and also [7], Remark 6.6). Theorem 2.9.…”

Locally conformally Kähler solvmanifolds: a survey

“…Indeed, it was shown in [71] that this 6-dimensional LCK manifold has first Betti number b 1 = 2 (more generally, b 1 = s for any OT manifold of type (s, t)). Using the description of the OT manifolds as solvmanifolds, Kasuya proved in [57] that the Betti numbers of an OT manifold of type (s, 1) are given by b p = b 2s+2−p = s p for 1 ≤ p ≤ s and b s+1 = 0. Therefore, for even s these manifolds also provide counterexamples to the Vaisman's conjecture.…”

“…Recalling the description of OT manifolds as solvmanifolds Γ\G, one may wonder if the isomorphisms (5) and (6) still hold for this class of solvmanifolds, even if G is not completely solvable. For the de Rham cohomology, it is shown in [57] that (5) holds in the case of OT manifolds of type (s, 1). For the Morse-Novikov cohomology, it was shown in [12] that the isomorphism (6) holds for a special class of OT manifolds of type (s, 1), namely, those satisfying the Mostow condition.…”

“…In [18] it was shown that any nilmanifold with an invariant complex structure has holomorphically trivial canonical bundle, while it is known that this does not happen generally for solvmanifolds (see [38]). Other results regarding complex geometric structures on nilmanifolds and solvmanifolds can be found in [8,29,57,62,93,94], among many others.…”

Rank two jump loci for solvmanifolds and Lie algebras

“…A solv-lattice is a discrete, co-compact subgroup Γ of a 1-connected solvable real Lie group S, giving rise to the compact, aspherical solvmanifold M " S{Γ, with fundamental group Γ. For such a manifold, Kasuya constructed in [14] a connected, finite-dimensional cdga model. Since the details are rather complicated, we are going to extract only the properties of the Kasuya model A of M that are relevant to our study, following [14,16].…”