Invariants of almost complex and almost Kähler manifolds

Abstract

Given a compact almost complex manifold [Formula: see text], the almost complex invariant [Formula: see text] is defined as the complex dimension of the cohomology space [Formula: see text]. Its properties have been studied mainly when [Formula: see text]. If we endow [Formula: see text] with an almost Hermitian metric [Formula: see text], then the number [Formula: see text], i.e. the complex dimension of the space of Hodge–de Rham harmonic [Formula: see text]-forms, does not depend on the choice of almost Kähler metrics when [Formula: see text]. In this paper, we study the relationship between [Formula: see text] and [Formula: see text] in dimension [Formula: see text]. We prove [Formula: see text] if [Formula: see text] is non-integrable and observe that [Formula: see text] if the metric is almost Kähler. If [Formula: see text] is a compact quotient of a completely solvable Lie group and [Formula: see text] is a left-invariant almost Kähler structure on [Formula: see text], we prove [Formula: see text]. Finally, we study the [Formula: see text]-pure and [Formula: see text]-full properties of [Formula: see text] on [Formula: see text]-forms for the special dimension [Formula: see text].