Hard Lefschetz Theorem for Sasakian manifolds

Abstract

We prove that on a compact Sasakian manifold $(M, \eta, g)$ of dimension $2n+1$, for any $0 \le p \le n$ 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.