1-Lefschetz contact solvmanifolds
Adrián Andrada, Agustín Garrone
TL;DR
The paper addresses the problem of characterizing $1$-Lefschetz contact solvmanifolds by establishing an equivalence: a contactization $(\mathfrak{g},\eta)$ of a unimodular symplectic Lie algebra $(\mathfrak{h},\omega)$ is $1$-Lefschetz iff $(\mathfrak{h},\omega)$ is $1$-Lefschetz. This is achieved through explicit cohomology relations between $\mathfrak{h}$ and its contactization, and by examining the interplay of commutator subalgebras under the central extension. The results yield a Benson–Gordon-type characterization in the contact setting, including a nilpotent case where a $1$-Lefschetz nilmanifold with invariant contact form must be a Heisenberg quotient, and enable construction of several examples of $1$-Lefschetz solvmanifolds, some of which are $2$-Lefschetz and others not. The work thus extends odd-dimensional Lefschetz theory to solvmanifolds via contactization, clarifying when Sasakian structures can exist and providing explicit lattices and families of manifolds illustrating the spectrum of Lefschetz behavior. These results deepen the understanding of cohomological Lefschetz properties in the contact realm and offer concrete building blocks for further geometric and topological investigations.
Abstract
We study the contact Lefschetz condition on compact contact solvmanifolds, as introduced by B.\ Cappelletti-Montano, A.\ De Nicola and I.\ Yudin. We seek to fill the gap in the literature concerning Benson-Gordon type results, characterizing $1$-Lefschetz contact solvmanifolds. We prove that the $1$-Lefschetz condition on Lie algebras is preserved via $1$-dimensional central extensions by a symplectic cocycle, thereby establishing that a unimodular symplectic Lie algebra $(\mathfrak{h}, ω)$ is $1$-Lefschetz if and only if its contactization $(\mathfrak{g}, η)$ is $1$-Lefschetz. We achieve this by showing an explicit relation for the relevant cohomology degrees of $\mathfrak{h}$ and $\mathfrak{g}$. Using this, we show how the commutators $[\mathfrak{h},\mathfrak{h}]$ and $[\mathfrak{g},\mathfrak{g}]$ are related, especially when the $1$-Lefschetz condition holds. By specializing to the nilpotent setting, we prove that $1$-Lefschetz contact nilmanifolds equipped with an invariant contact form are quotients of a Heisenberg group, and deduce that there are many examples of compact $K$-contact solvmanifolds not admitting compatible Sasakian structures. We also construct examples of completely solvable $1$-Lefschetz solvmanifolds, some having the $2$-Lefschetz property and some failing it.