Hard Lefschetz theorem for Sasakian manifolds

Abstract

We prove that on a compact Sasakian manifold $(M,\eta, g)$ of dimension $2n + 1$, for any $0 \leq p \leq n$ the wedge product with $\eta \wedge (d \eta)^p$ defines an isomorphism between the spaces of harmonic forms $\Omega^{n-p}_{\Delta}(M)$ and $\Omega^{n+p+1}_{\Delta}(M)$. Therefore it induces an isomorphism between the de Rham cohomology spaces $H^{n-p}(M)$ and $H^{n+p+1}(M)$. Such isomorphism is proven to be independent of the choice of a compatible Sasakian metric on a given contact manifold. As a consequence, an obstruction for a contact manifold to admit Sasakian structures is found.