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 ${\Omega }_{\Delta }^{n-p}(M)$ and ${\Omega }_{\Delta }^{n+p+1}(M)$. 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.

19 pages, 1 figure, accepted for publication in the Journal of Differential Geometry

References in corpus (3)

• Canonical Sasakian Metrics
• Lefschetz contact manifolds and odd dimensional symplectic geometry
• Extremal Sasakian horizons

Cited by in corpus (1)

• Homology Smale-Barden manifolds with K-contact and Sasakian structures