Hard Lefschetz Theorem for Sasakian manifolds

TL;DR

This paper proves a Lefschetz-type isomorphism for harmonic forms on compact Sasakian manifolds, establishing a deep link between geometric structures and cohomology, and identifying obstructions to Sasakian structures on contact manifolds.

Contribution

It establishes a Lefschetz theorem for harmonic forms on Sasakian manifolds and identifies obstructions for contact manifolds to admit Sasakian structures.

Findings

Isomorphism between harmonic forms via wedge product with η and dη

Independence of the isomorphism from the choice of Sasakian metric

Obstruction criterion for contact manifolds to admit Sasakian structures

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.