The Hard Lefschetz Theorem on Kähler Lie Algebroids

Abstract

Compact Kähler manifolds classically satisfy the Hard Lefschetz Theorem, which gives strong control on the underlying topology of the manifold. One expects a similar theorem to be true for Kähler Lie Algebroids, and we show for a certain class of them that this is indeed true, with an added ellipticity requirement. We provide examples of Lie Algebroids satisfying this, as well as an example of a Kähler Lie Algebroid that does not meet this Ellipticity requirement, and consequently fails to satisfy the Hard Lefschetz condition.