Coeffective basic cohomologies of $K$--contact and Sasakian manifolds

TL;DR

This paper introduces and studies coeffective basic cohomologies on $K$--contact and Sasakian manifolds, establishing relations with classical basic cohomology and proving Hodge decomposition theorems.

Contribution

It defines coeffective basic cohomologies for $K$--contact and Sasakian manifolds and explores their properties and relations with existing cohomologies.

Findings

Inequalities relating coeffective and Betti numbers for finite type manifolds.

Definition and analysis of coeffective Dolbeault and Bott-Chern cohomologies.

Hodge decomposition theorems linking coeffective basic cohomologies.

Abstract

In this paper we define coeffective de Rham cohomology for basic forms on a $K$--contact or Sasakian manifold $M$ and we discuss its relation with usually basic cohomology of $M$. When $M$ is of finite type (for instance it is compact) several inequalities relating some basic coeffective numbers to classical basic Betti numbers of $M$ are obtained. In the case of Sasakian manifolds, we define and study coeffective Dolbeault and Bott-Chern cohomologies for basic forms. Also, in this case, we prove some Hodge decomposition theorems for coeffective basic de Rham cohomology, relating this cohomology with coeffective basic Dolbeault or Bott-Chern cohomology. The notions are introduced in a similar manner with the case of symplectic and K\"ahler manifolds.